The CCW don'ts are here:
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.
Actually the emboldened part is a bit misleading, see RSMO 571.107.1(15). If the location is posted you're not supposed to carry inside. Indicating that "basically nothing is off-limits" is also misleading. There are 18 specific locations, including those covered by Federal law (item #9), that are no-carry authorized areas and one of those 18 IS NOT listed in the same statute as the other 17; that would be the public bus. A public bus is listed under Missouri's Bus Hijacking statute. Posted property is also one of those 17 items; best bet is to not patronize those posted places.
RSMO 571.101.2 is very relevant and very important, especially since some LEOs in this State refuse to properly educate themselves to this fact and/or their departments refuse to properly train them on this. Here is the first part of Subsection 2 and it's relevant, you can be charged with Trespass if you refuse to leave, but you likely know that already. "Carrying of a concealed firearm in a location specified in subdivisions (1) to (17) of subsection 1 of this section by any individual who holds a concealed carry endorsement issued pursuant to sections 571.101 to 571.121 shall not be a criminal act but may subject the person to denial to the premises or removal from the premises. If such person refuses to leave the premises and a peace officer is summoned, such person may be issued a citation for an amount not to exceed one hundred dollars for the first offense. If a second citation for a similar violation occurs within a six-month period, such person shall be fined an amount not to exceed two hundred dollars and his or her endorsement to carry concealed firearms shall be suspended for a period of one year. If a third citation for a similar violation is issued within one year of the first citation, such person shall be fined an amount not to exceed five hundred dollars and shall have his or her concealed carry endorsement revoked and such person shall not be eligible for a concealed carry endorsement for a period of three years."
http://www.moga.mo.gov/statutes/C500-599/5710000107.HTM
Why it is so relevant for a visitor to Missouri is because once you're convicted in Missouri your record DOES NOT go away, ever. It illustrates what Missouri Statute calls "a pattern of behavior" that could be passed to the CCW permit holder's issuing State and we all know the can of worms that can open.
DO NOT attempt to use handgunlaw.us as anything more than a potential source of information. It IS NOT and CANNOT be used for your legal defense if you manage to be in violation of the law by going on their information, their disclosure notice says it's for entertainment purposes only. Go straight to the Revised Missouri Statutes and print out what you need, study it a bit, and carry a copy of the applicable statutes with you. Consider the printed statutes like having a pocket-lawyer along for the ride.
RSMO 571 is the weapons statute, including the CCW stuff.
RSMO 563 is the Defense Justifications Statute (very important!)
http://www.moga.mo.gov/statutes/c571.htm
http://www.moga.mo.gov/statutes/c563.htm
Should you need it, here's the MOHP's own brochure...just remember that the printed law trumps this, it's informational only.
http://www.mshp.dps.mo.gov/MSHPWeb/Publications/Brochures/documents/SHP-863.pdf