"But if it's just using the physical plant then it's different. What makes an idol an idol is how those seeing it feel about it. I am a Christian, but I don't let statues of religious significance to other religions bother me. It's only an idol if I choose to treat it as one. Otherwise it's just another pretty statue, or maybe not depending on what the statue is."
I agree with you, personally.
But the prohibition of Orthodox Jews from entering into a place of Christian worship was halichially decided (to vastly oversimply, by the Sanhedrin), and personal feelings aside, 'tis the Law for Jews (until such time as a comprable or higher "court" makes a different ruling) that entry would be a endorsement of polytheism, and thus violate the rule of having no God, but God.
Again, have no clue about muzzies, and doubt it is a true objection, as I have seen many muslims enter Churches.
Most large churches have meeting spaces other than the main sanctuary -- classrooms, meeting rooms used for everyone from scouts to community bands, even gymnasiums. Those churches double as community centers, and there's usually a pretty clear line between the sacred spaces and the ones devoted to community functions.
I've voted in several polling places at churches, but I've never seen the voting booths set up in the sanctuary. It's been my impression that both the church and the state are comfortable with this distinction.