http://www.moga.mo.gov/statutes/c500-599/5710000107.htm Basically, there is nothing off limit, except federal facilities. But, if you are asked to leave it would be a really good idea to do so. If the LEO's are called, you can be hit with a hefty fine.
You are not obligated to inform
You may carry in your car as much as you like, if you are over 21 and it's a handgun: http://www.moga.mo.gov/statutes/c500-599/5710000030.htm Section 3.
No GFZ posting requirements, watch your step.
There's really no off limit locations (except federal), but I'd definitely obey any signs.