1) You're wrong. No foreign national (even resident aliens) has the right to possess a weapon on US soil. Use FindLaw to see the relevant federal statute. So, NO, the Mexican army cannot march into Tucson under arms even if they don't fire a shot. Nor can any illegal alien of any nationality.
2) Voting rights are indeed the territory of the states and federal laws, but they are also governed in the amendments of the constitution, specifically the 14th. In that wondrous amendment, it says that the equal protection of the laws shall apply to all citizens. So, if voting is allocated to citizens, it must be allocated to all citizens. The 14th does not extend to NON-citizens (Dershowitz and the ACLU disagree with me on this).
3) "Congress shall make no law respecting an establishment of religion" (from memory). Didn't say nuthin' about the religion of an individual. It said about congress making laws about religions in general.
4) Can Congress make a law depriving non-citizens of jury trials? Absolutely, and they have. The INS routinely conducts trials which adjudicate whether an alien is allowed to be in the United States or not. That's a matter of criminal law, and yet there is no 'jury of his peers' present. The fact that aliens get tried in our courts all the time is a matter that has thousands of pages of case history, and I won't try to challenge all of it, because its gone both ways.