I'm confused about the American Republican party

[kandrathe]

That sort of blew my mind because with my understanding of the 1st amendment, church leaders should never have influence over politics - right?

You misunderstand the 1st amendment. The bill of rights protects the citizens from the government. So the (Federal) government cannot establish a national religion, but it's up to each state to determine how it wants to operate. In fact, many states did have a state religion early in our history.

As of 2010, Article III of the Massachusetts constitution still provided, "... the legislature shall, from time to time, authorize and require, the several towns, parishes, precincts, and other bodies politic, or religious societies, to make suitable provision, at their own expense, for the institution of the public worship of God, and for the support and maintenance of public Protestant teachers of piety, religion and morality, in all cases where such provision shall not be made voluntarily."

Since "who is elected" has a great impact on policy that may go against religious beliefs, religious leaders are obligated to engage in forming and sharing their political opinions. Many don't. But, some powerful ones do. For example, the lastest issue with the Obama administration forcing Catholic religious organizations to provide insurance coverage for contraception, when it is against their religious beliefs. Even liberal Catholics are upset, because they sought and got specific assurances that within the rule making phase of rolling out this legislation this confrontation would not happen. Evangelicals are mostly united against abortion, against changing the sacrement of marriage, and against "smut" on the airwaves/cable.

So... back to the 1st amendment. As a citizen, you are (or should be) entirely protected from government interference in practicing your religious beliefs. But, the federal government is prevented from promoting, advocating, or establishing any type of religious act (other than the ones they have -- and no one has yet to win a case against e.g. our money, our pledge of allegiance, etc. ).

The recent SCOTUS decision Citizens United, especially emphasized the defense of political speech... "If the First Amendment has any force, it prohibits Congress from fining or jailing citizens, or associations of citizens, for simply engaging in political speech."