Separation of Church and State

We claim to separate church and state but we really don't.

Many parts of our state, beyond having their foundations in religious beliefs, have direct religious references. Our currency states "In God We Trust". Our state governments require citizens to get a marriage license. Our pledge of allegiance mentions God. The entire country operates around Christian holidays.

We did specifically forbid the government from promoting or favoring one religion over another, with the exception of the de-facto endorsement of Christianity, but that's more like a limitation rather than a separation.

If we wanted to construct a government that was truly separate from religion, how would we do it?

The main issue that comes to my mind is crime.  Governments create laws and people who break the laws must be punished. But is there a difference between committing a crime and breaking a law? Or between committing a crime and sinning? Is there such a thing as a victim-less crime? Are some acts crimes in one religion but not another? Is it fair for the government to pass laws that essentially echo the laws of a specific favored religion?

I think that because religions regulate behavior, it's not possible to construct a state that is completely separate from religion - the state needs to either adopt a specific religion (or atheism) and align its laws with it, or it needs to maintain a minimal set of laws to set a basic standard of behavior and then regulate the religions that can operate within this minimal framework.

Should religious institutions have the authority to punish their members for acts that violate the laws of their religion? What maximum punishment will the state allow?

What should happen if a member of one religion commits an act that hurts a member of another religion in some way, and the two religions have different positions on the act? What if the perpetrator's religion declares it allowed and the victim's religion declares it is not allowed? What if the opposite happens - the perpetrator's religion allows the action but the victim's religion forbids it? Should the state handle these cases or should there be a regulation on how they are handled and how disagreements will be resolved?