Here are some general and specific tips focused around Rogues staying alive.
Gear:
Agility helps. It gives you dodge in addition to its offensive power. I don't find +dodge% bonuses worth it, as dodge is too costly compared to the offensive power you can get, but there are some times where dodge will save you.
Stamina is the rule, though. The biggest way for a Rogue to survive is simply to buy himself time. If you're dying too often, check your stamina first. Dodge may help you against physical attacks, but it's luck based. It also doesn't help you against spells.
Rogue Skills:
Cheap Shot / Kidney Shot / Gouge -> Something doing no damage is doing no damage to you. If you're in a really tight spot, trying to get out of a Mob's range or waiting for Feint to come up, one of these might give you just enough time to get away safely.
Feint -> If you don't have aggro, you're taking a lot less damage ;)
Sprint -> Only useful if you're trying to get out of a mob's range in time.
Kick -> Caster-only solution, but as part of your interrupt arsenal, you can render many casters useless.
Evasion -> You shouldn't rely on Evasion for survival, unless all other options are burned. Use Evasion when you think you'll take damage, but want to be safer, or if you're just burning through mobs and are trying to cutdown on downtime. It's a help, but it's not really a solution.
Blind -> Similar to the stuns, it's useful in the "give me 2 more seconds" sense. It's reagent cost means it shouldn't be your first choice, though.
Distract -> Never underestimate distract, in any situation. Between splitting pulls, helping set up a Sap (specifically in PvP), preventing (or delay) adds, Distract has tons of uses. If something is not in combat but is coming towards you, remember Distract ;)
Vanish -> The big one. This one will save your life more times than you can count. Use it at the highest level, and it will save you even more.
No DoTs: Simply vanish. You win, unless something sees through stealth.
Trapped: Don't forget Vanish breaks entrapments! Sometimes using Vanish on Frost Nova is just a waste of a reagent. Other times, it turns a lose into a win by itself.
DoTs: Advanced Vanish usage. While you have DoTs on you, Vanish can still serve a purpose. If you're fighting a mob and you moved it from where the fight started, a Vanish should cause it to go into Evade mode and head back, buying you time to run away. Even if it's two yards, it may save your life. Another use (mainly PvP, due to evade-reset) is time time your DoTs. Right after the DoT procs, Vanish. Quickly hit your opening move, cheap shot being the best since you're in trouble theoretically, and you just got an opening move off while still under DoT. The key here is that Vanish/Stealth isn't prevented from DoTs. The DoT's damage is what breaks stealth.
First Aid: Always have Heavy Runecloth Bandages ready. Basically, if you don't have a DoT and are missing more than 2k health, it's a good time to bandage.
Specific Fights:
Zul'Gurub:
Exploding Bats -> If you (under no lag), respond to the notice immediately, you should not die. Do not back up, either strafe or run forward. If there's more than one exploding bat, I leave the first one at 40% all the time and head to the second one early. If dying is still a problem, simply back off early in every fight. The ranged DPS can handle these guys.
Jeklik -> Fire Resistance helps here. Getting hit by 3 bombs while in some/all FR gear is a lot less deadly than 3 bombs in no FR gear.
Marli -> Bandages will save you here. Run away from the poison, wait out the DoT, and bandage. Run back in.
Venoxis -> Fast reaction times are simply the best solution here. That poison cloud hurts. If bandage is not ready and I'm iffy on health, sometimes I'll just jump back into the fight. If I die, I die. If not, well I helped kill him faster.
Mandokir/Arlokk -> Watch the spinning. The instant they spin, strafe. You won't get out of the first whirlwind, but you may get out of the second. This is probably the single best use for dodge PvE Raids, reducing damage from the whirlwinds. Even better, have a mage use Detect Magic on the mob. The Whirlwind icon comes up before they start spinning, buying you a tiny bit of extra time.
Onyxia: To borrow from Aedak: "80. Stay... away from the dragon's face. The breath comes out of its face.
81. Stay... away from the drapon's tail. It probably won't kill you, but it sure does complicate things later." I'll vanish at around 85% in Phase 1 and around 20% in Phase 2 to prevent dying from aggro :)
Molten Core:
Lava Packs -> FR rules your survivability, due to random Fire Blossoms and AoEs from both Flameguard and Firewalker. If you get too many Incite Flames on you, simply walk away.
Lucifron -> If you don't have 4k hitpoints, you're really hurting yourself here. You can't survive 2 of Lucifron's magic debuffs without resists/heals/bandages. Bandage when you can. If you are under 2k hitpoints, the corner of the cave is your safe point. Assuming Lucifron's in the right spot, he won't hit you with the debuff from there.
Magmadar -> Again, FR is your savior. The Magma Spits add up, the breath is almost always going to hit you, and bad luck with loogies takes up any "spare" health you had. Make sure to save your bandages for when you have no Magma Spits on you.
Gehennas -> I don't go too much FR here, but some can help if you keep getting under Rain of Fire.
Garr to Golemagg -> nothing really any different from any other class.
Majordomo -> Some FR/SR can help here. When you Bandage, try to jump over the tiny ledge onto the rise to get to Domo, the LoS will help make sure you don't get hit with a random AoE or Shadowbolt while bandaging.
Ragnaros -> Pure FR here. When Ragnaros is engaged, start counting. His AoE Knockback is ~every 25-30 seconds, but the first timer from CTRA will be wrong. Be ready. After the Sons of Flame phase, the first AoE Knockback is somewhat random. Wait for it to reengage. At that point CTRA will be messed up again, so make sure to count yourself between the first and second attacks.
The AoE Knockback seems to be a complete aggro wipe. Because if this, I usually Feint once early after a Knockback and then go all out until the next round of backing up. If you get aggro against Ragnaros, get out of his range immediately. He'll switch back to highest aggro in range target. Wait a few seconds (and probably some heals) before going back in, or you'll still be top aggro.
As a final note, Ragnaros is the only fight I actually back up to get away from something. In every other fight, I strafe.
Blackwing Lair (lacking on my part):
Razorgore -> For Phase 1, stuns+gouge are the way I survive versus mages. Kick if they're trying to fireball. Bandage in between spawns, but beware that bandaging may get you heal aggro. If you get heal aggro, you can Vanish once. After that, you're a kiter. For Phase 2, honestly, be prepared to die. Pull every trick you have here, a Fireball Volley + Warstomp + Cleave can truly hurt you. After you've used every trick, you'll typically go back in to simply lose your health again.
Vaelestrasz -> Same as Onyxia. Vanish after first tank transition to drop your aggro.
Suppression Room -> Full FR is great here with the dragonkin. If you get around 6 debuffs stacked, run away and wait it out, then run back in. It only lasts 5 seconds, but it certainly hurts if you let it stack too much. Don't worry about the whelps, they aren't your responsibility.
Broodlord -> FR for the Blast Wave, which also has the benefit of lowering your threat due to less damage. I'm not sure where the line is, due to his knockback deaggros, but it's fairly safe to say you shouldn't try treading it. Feint throughout is good here, and a Vanish later on in the fight, when the tanks are nearing their threat ceiling due to the knockback, is a good help. You can simply walk away and bandage when low from his AoE.
Firemaw -> FR, timing, and bandages. The higher your FR, the less often you have to run out of LoS because of his Flame Buffet. I seemed to reach a sweetspot around 270 FR last night, where every time I ran away my bandages were almost ready.
Best angle (assuming MT in front and OTs directly behind), is right to the side of Firemaw, slightly angled to the back.
Firemaw's Wing Buffet and Shadow Flame require different reactions depending on the timing:
Shadow Flame -> Do nothing. You're fine. If you're not fine, tweak your positioning.
Wing Buffet -> Swap to slightly angled to the front.
Both -> Get the hell outta dodge. You don't know which way he'll be facing when the Shadow Flame hits, so don't try your luck.
Gear:
Agility helps. It gives you dodge in addition to its offensive power. I don't find +dodge% bonuses worth it, as dodge is too costly compared to the offensive power you can get, but there are some times where dodge will save you.
Stamina is the rule, though. The biggest way for a Rogue to survive is simply to buy himself time. If you're dying too often, check your stamina first. Dodge may help you against physical attacks, but it's luck based. It also doesn't help you against spells.
Rogue Skills:
Cheap Shot / Kidney Shot / Gouge -> Something doing no damage is doing no damage to you. If you're in a really tight spot, trying to get out of a Mob's range or waiting for Feint to come up, one of these might give you just enough time to get away safely.
Feint -> If you don't have aggro, you're taking a lot less damage ;)
Sprint -> Only useful if you're trying to get out of a mob's range in time.
Kick -> Caster-only solution, but as part of your interrupt arsenal, you can render many casters useless.
Evasion -> You shouldn't rely on Evasion for survival, unless all other options are burned. Use Evasion when you think you'll take damage, but want to be safer, or if you're just burning through mobs and are trying to cutdown on downtime. It's a help, but it's not really a solution.
Blind -> Similar to the stuns, it's useful in the "give me 2 more seconds" sense. It's reagent cost means it shouldn't be your first choice, though.
Distract -> Never underestimate distract, in any situation. Between splitting pulls, helping set up a Sap (specifically in PvP), preventing (or delay) adds, Distract has tons of uses. If something is not in combat but is coming towards you, remember Distract ;)
Vanish -> The big one. This one will save your life more times than you can count. Use it at the highest level, and it will save you even more.
No DoTs: Simply vanish. You win, unless something sees through stealth.
Trapped: Don't forget Vanish breaks entrapments! Sometimes using Vanish on Frost Nova is just a waste of a reagent. Other times, it turns a lose into a win by itself.
DoTs: Advanced Vanish usage. While you have DoTs on you, Vanish can still serve a purpose. If you're fighting a mob and you moved it from where the fight started, a Vanish should cause it to go into Evade mode and head back, buying you time to run away. Even if it's two yards, it may save your life. Another use (mainly PvP, due to evade-reset) is time time your DoTs. Right after the DoT procs, Vanish. Quickly hit your opening move, cheap shot being the best since you're in trouble theoretically, and you just got an opening move off while still under DoT. The key here is that Vanish/Stealth isn't prevented from DoTs. The DoT's damage is what breaks stealth.
First Aid: Always have Heavy Runecloth Bandages ready. Basically, if you don't have a DoT and are missing more than 2k health, it's a good time to bandage.
Specific Fights:
Zul'Gurub:
Exploding Bats -> If you (under no lag), respond to the notice immediately, you should not die. Do not back up, either strafe or run forward. If there's more than one exploding bat, I leave the first one at 40% all the time and head to the second one early. If dying is still a problem, simply back off early in every fight. The ranged DPS can handle these guys.
Jeklik -> Fire Resistance helps here. Getting hit by 3 bombs while in some/all FR gear is a lot less deadly than 3 bombs in no FR gear.
Marli -> Bandages will save you here. Run away from the poison, wait out the DoT, and bandage. Run back in.
Venoxis -> Fast reaction times are simply the best solution here. That poison cloud hurts. If bandage is not ready and I'm iffy on health, sometimes I'll just jump back into the fight. If I die, I die. If not, well I helped kill him faster.
Mandokir/Arlokk -> Watch the spinning. The instant they spin, strafe. You won't get out of the first whirlwind, but you may get out of the second. This is probably the single best use for dodge PvE Raids, reducing damage from the whirlwinds. Even better, have a mage use Detect Magic on the mob. The Whirlwind icon comes up before they start spinning, buying you a tiny bit of extra time.
Onyxia: To borrow from Aedak: "80. Stay... away from the dragon's face. The breath comes out of its face.
81. Stay... away from the drapon's tail. It probably won't kill you, but it sure does complicate things later." I'll vanish at around 85% in Phase 1 and around 20% in Phase 2 to prevent dying from aggro :)
Molten Core:
Lava Packs -> FR rules your survivability, due to random Fire Blossoms and AoEs from both Flameguard and Firewalker. If you get too many Incite Flames on you, simply walk away.
Lucifron -> If you don't have 4k hitpoints, you're really hurting yourself here. You can't survive 2 of Lucifron's magic debuffs without resists/heals/bandages. Bandage when you can. If you are under 2k hitpoints, the corner of the cave is your safe point. Assuming Lucifron's in the right spot, he won't hit you with the debuff from there.
Magmadar -> Again, FR is your savior. The Magma Spits add up, the breath is almost always going to hit you, and bad luck with loogies takes up any "spare" health you had. Make sure to save your bandages for when you have no Magma Spits on you.
Gehennas -> I don't go too much FR here, but some can help if you keep getting under Rain of Fire.
Garr to Golemagg -> nothing really any different from any other class.
Majordomo -> Some FR/SR can help here. When you Bandage, try to jump over the tiny ledge onto the rise to get to Domo, the LoS will help make sure you don't get hit with a random AoE or Shadowbolt while bandaging.
Ragnaros -> Pure FR here. When Ragnaros is engaged, start counting. His AoE Knockback is ~every 25-30 seconds, but the first timer from CTRA will be wrong. Be ready. After the Sons of Flame phase, the first AoE Knockback is somewhat random. Wait for it to reengage. At that point CTRA will be messed up again, so make sure to count yourself between the first and second attacks.
The AoE Knockback seems to be a complete aggro wipe. Because if this, I usually Feint once early after a Knockback and then go all out until the next round of backing up. If you get aggro against Ragnaros, get out of his range immediately. He'll switch back to highest aggro in range target. Wait a few seconds (and probably some heals) before going back in, or you'll still be top aggro.
As a final note, Ragnaros is the only fight I actually back up to get away from something. In every other fight, I strafe.
Blackwing Lair (lacking on my part):
Razorgore -> For Phase 1, stuns+gouge are the way I survive versus mages. Kick if they're trying to fireball. Bandage in between spawns, but beware that bandaging may get you heal aggro. If you get heal aggro, you can Vanish once. After that, you're a kiter. For Phase 2, honestly, be prepared to die. Pull every trick you have here, a Fireball Volley + Warstomp + Cleave can truly hurt you. After you've used every trick, you'll typically go back in to simply lose your health again.
Vaelestrasz -> Same as Onyxia. Vanish after first tank transition to drop your aggro.
Suppression Room -> Full FR is great here with the dragonkin. If you get around 6 debuffs stacked, run away and wait it out, then run back in. It only lasts 5 seconds, but it certainly hurts if you let it stack too much. Don't worry about the whelps, they aren't your responsibility.
Broodlord -> FR for the Blast Wave, which also has the benefit of lowering your threat due to less damage. I'm not sure where the line is, due to his knockback deaggros, but it's fairly safe to say you shouldn't try treading it. Feint throughout is good here, and a Vanish later on in the fight, when the tanks are nearing their threat ceiling due to the knockback, is a good help. You can simply walk away and bandage when low from his AoE.
Firemaw -> FR, timing, and bandages. The higher your FR, the less often you have to run out of LoS because of his Flame Buffet. I seemed to reach a sweetspot around 270 FR last night, where every time I ran away my bandages were almost ready.
Best angle (assuming MT in front and OTs directly behind), is right to the side of Firemaw, slightly angled to the back.
Firemaw's Wing Buffet and Shadow Flame require different reactions depending on the timing:
Shadow Flame -> Do nothing. You're fine. If you're not fine, tweak your positioning.
Wing Buffet -> Swap to slightly angled to the front.
Both -> Get the hell outta dodge. You don't know which way he'll be facing when the Shadow Flame hits, so don't try your luck.
Trade yourself in for the perfect one. No one needs to know that you feel you've been ruined!