There are more knowledgeable people than I that should come along but I will make a stab at your questions.
OC is legal state wide BUT be forewarned that by state law the local municipalities have the right to have anti open carry ordinances enacted. Some do and some don't so it is quite a patchwork that can get a person in trouble rather quickly.
No firearms registration in MO.
Private face to face sales are legal as long as the persons are lawful to purchase.
Concealed carry in a vehicle is allowed with a concealable weapon without a concealed carry permit but do not step out of the vehicle with it concealed on you. without a CCW permit.
You may apply for a CCW permit in MO as soon as you establish residency with proof of such. We no longer have the silly 6 month residency wait to apply.
Hope this helps.