Is there a recommended distance between fasteners on sole plates?
I'm assuming none of these are structural shear walls since shot pins are inappropriate for such use. There is some arbitrary lateral load that interior partitions are supposed to resist, it's only 5 or 10 psf from memory, I can't find the reference at the moment. I'm not sure what the shot pins are rated for either. So without any more specific data, I would judge that a 4 foot spacing should be about right. There's really no need for doubled fasteners at the ends, as long as there's at least one good shot at the ends. You do not want to place shots too close together, the second shot weakens the first.

There is some minimum spacing determined by the shot pin manufacturer to achieve a certain rated strength. This is usually outlined in so called "ICBO Reports" or similar. I would guess a 5-6" minimum spacing would be safe. The only reason to place shots this close is if the first one spalled out the concrete.