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.