Which is why I say FUCK RELIGION.

The first amendment to the constitution actually as two key clauses dealing with religion, one prohibiting laws that prevent people from practicing their religion, and one prohibiting laws that promote or respect any kind of religion.
"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof"
But to address those why the US has so much crazy shit that goes on in politics due to religious influence, there are basically two reasons behind that. First, while the Constitution basically says the government needs to keep its hands completely out of religion, separating the individuals that make up government from whatever personal religious beliefs that they hold isn't so easy, and very few politicians have the kind of character required to set aside their religious beliefs when making decisions regarding laws. The second reason is that government aside religious beliefs are very deeply entrenched in the populace here, and to add to that there's a sizable enough contingent of batshit crazy individuals (aka the "religious right") that are very vocal and active in politics, and enough of the rest of the populace often just go along with whatever they propose. Also as part of this deep religious entrenchment there's generally resistance against secularly dealing with anything that has historically had religious ties, and often political backlash against anyone who tries to advocate for secularly handling such matters (openly stating that one doesn't believe in god is typically a surefire way to prevent oneself from ever getting elected to any major public office).

Heh...this reminds me of this news (It's about several jurors who consulted the Bible to decide a death penalty).

Every man must act according to his beliefs (presumably the same applies to women as well) but the lack of character thing is the problem. When officials forget that their election is a vote of support for their governing ability rather than for their personal beliefs, you're bound to get a pinheaded prick forcing his personal beliefs on the people he supposedly serves even when informed that said beliefs are in direct opposition to the will of the very people he serves.
Yes, I'm looking at YOU Atkinson!

Yes. The government cannot stop people from worshipping their chosen deity. Nor can they impose religious practices upon them. Yet that seems only to have had minimal effect in practise. The first amendment doesn't stop religion from influencing politics. Either in its wording or practice.
Take abortion and Bush's 'moral' stance on it. Or his funding of religous organisations. Which had no religious influence at all, of course. :P Or the fact that homosexual couples are still so far from having equal rights. Again, like you point out there is a heavy religious influence there.
Essentially, with America being both Christian and somewhat democratic religious influence is completely unavoidable.

A fuck religion might be pretty sweet. I think they call that hedonism.

Then I must be a very hedonistic person. >=)