As stated, there are roles in government that must be separated from religion just as there are roles in religion that must be separated from government. Yet I think two questions actually get subsumed in this one. The second question is about government's role in relation to individuals' personal religion.
For instance, should a President be able, without judgement or stricture, to call a national day of thanksgiving, as Washington and Lincoln did? Yes. Absolutely a President should be able to do this. Should a school be able to restrict religious students from practicing publicly suitable elements of their religion on campus? No. Absolutely a school must not do this. At least some of our ideas today about government and religion are really about government and individuals' religion. So while government and religion work well in such a diversity society when they are "hands off" each other, there is a fine fine line that divides church and state from individual religion and state.
Of course, this can get very tricky. When I moved from California's Central Valley some fifteen years ago, there was a controversy charted to come before the courts about Southeast Asian religious/shamanic practices that required small animal sacrifices, the remains of which were found inside mystical symbols inscribed in the dirt of backyard alleys. I never found out how this issue was resolved as it went to court years after I had moved, but this illustrates the difficulty of any discussion about religion, which encompasses many facets, personal religion, and government.
I think that government and religion should be totally separate in our government because that is consistent with the First Amendment to the US Constitution and its interpretation by our courts. However, I think it would be arrogant of me to think that this would work in all nations, under all circumstances. In a very homogeneous society, if it is the will of the people to have a theocratic form of government, I don't see how the opinion of people with other forms of government should matter. Our pluralistic culture and our Constitution accustom us to this concept, and thus far, it has worked well for us.
I do take note of frequent references in politics today to our being a "Christian nation," and that makes me nervous because it suggests there is a contingency of people who do want to "establish" Christianity as our nation's religion, controlling decisions that should be personal and private. I am Jewish, and whenever I hear that phrase, I protest it. If this is a Christian nation, how can it be my nation, then, or the nation of people who are Sikhs, or Hindus, or Muslims, or Buddhists?
I'd agree with #4, the government and religion should be totally separate. The involvement of the government in providing any form of support to religious activities of any sort always creates conflict. Restricting the discussion to democracies, there is no nation where everyone practices the same religion. If the government were to provide any favors to those practicing a particular religion, it is totally justified for those practicing other religions to demand the same.
This raises the basic question about what governments are created for. Religion may be a very important aspect in the lives of a large number of people but does this make it the government's duty to provide funds to build churches or mosques or temples, or for the government to help in propagating their religion in any other way.
What is required of the government is to ensure that everyone has the freedom to choose what they would like to believe in and that nothing is being forced on anyone in any way.
This depends on what you would call a "total separation." The Establishment Clause says that government can't establish a religion, but there should be times when governments do things that could be seen as promoting religion. For example, it seems reasonable to allow government money to be used to bus students to private schools that are run by church groups. It makes sense that religious organizations should be allowed to bid to do things like providing government services so long as they do not proselytize to the people who receive those services.
There does not need to be a total separation. However, it should be fairly stringent so we do not return to the situation where we regularly have things like state sponsored prayer in schools.
My class has been readingAnthemby Ayn Rand, so I've been watching some interviews she did while she was alive (off of Youtube). It is interesting to note that Rand says that peace came when churches were declared separate from government. When organizations are free to grow at their own choosing, the world is better for it. So, if Rand were here, she'd probably say that bringing religion back to the government would only cause strife. Religions are the "real collectivists" so that wouldn't do in government. Plus, each religion wants to baptize others, so there would be a battle between which religion would dominate and that totally goes against the constitution. It wouldn't work to combine the two.
I agree with #3 in that there are areas where the church might be able to fill a services function that the government cannot fill, so they ought to be allowed to fulfill that "quasi-governmental" function. For example, the budget crisis is probably going to lead to a decrease in funding for the social "safety net." This is where churches could step in and provide a valuable humanitarian and charitable service. The government should be able to help churches do this, with the obvious restrictions on promoting or establishing a state-run religion.
While I do not think that the church should have too much influence in the government, and the government should not involve itself too much in the church, I do not think a total separation exists. In courts witnesses swear on a Bible. Some church groups get government money.