The process used to pick a President is only used for that election, the other officials are elected in a pretty straightforward method. House of Reps voted on every 2 years for 2 year terms, Senators have 6 year terms. The whole process of picking who runs for each party is long, drawn out, and possibly a little odd, but it is the way it is.
To tackle the religious question. I will say up front I am going to be pretty open with my personal views and do not mean to offend or convert anyone, just give my my gut(no pun intended given the state of my gut) feeling about religion and politics in the USA.
Every once in a while they do a poll asking people if they believe in God, generally about 80-90% say yes. Don't know how that would be answered in the UK, but I suspect perhaps not that high. Perhaps religion is so important here because so many of our forefathers came to this country to begin with for religious reasons, that seems to me to be the most logical reason. People with strong religious beliefs, strong enough to make them leave their homes, would probably raise families with strong beliefs. In my own case, I believe in God without question, but I do not regularly attend church, although my wife and parents do(in fact, just about everyone in my family out to my cousins attend church more than I do). While I do not attend church, I would never vote for someone for President who openly said they did not believe in God, and a great many of my countrymen are the same way. The reason behind this is fairly simple. I want a leader who shares my values system, because I want a country that reflects my values system. Perhaps that is arrogant, short sighted, what have you, but it is my feeling, and I think it is shared by most voters whether or not they say it. That is in a broad sense. The candidate does not have to think exactly the way I do, I am not hung up on the little things in his beliefs, but in the great scheme of things, I think it is hard to have good values unless they are based somewhat on a belief in God, and I don't trust someone who does not believe in God at all to lead this country. It seems obvious to me that if a great many others didn't share this view, religion wouldn't be a factor in our elections.
Now, having said that, I think that European news organizations overplay the impact religion has on our elections because Europe is so secular, and it seem primitive that we are not(in their opinion). So probably many of you believe religion plays a bigger part than it really does. At the end of the day, at least this day, if you don't publicly state you believe in God in the USA you are not going to get elected President. But that is about all you have to do, after the voting public is comfortable you are not a "heretic", we care most about the issues at hand. I bet the replies to this one will be interesting.