I dont know
In general, no. Most gun dealerships have policies that prohibit them from questioning their customers.
So do illegal ones, actually. There was a guy I knew who sold a gun to a man illegally. The man's son later used it to shoot up a school. That guy felt horrible until the day he died.
They probably care, because then they do hold some responsibility when firearms can be traced back to dealers. However, if dealers fulfill all their legal responsibilities, there really isn't much they can do.
Yes. Most places have policies in place that require them to grill customers and run background checks. On top of that, the smaller more private businesses are almost always run by a gun lover with an agenda. They like to feel they are supplying fellow NRA members and not criminals.