In GTA 5 story mode, the health is not regenerating

This could as already stated be a sign of a hardware issue in the PS3 itself, However don't overlook the possibility of it being a television problem. Try hooking up to another tv, and if that does not fix the problem there may even be something up with the AV or HDMI cable you are using. if swapping out these things does not help and you are still under a warranty I would recommend downloading all your save files to flash drives and return the ps3 for a new one.

as for the heat problem try to keep your PS3 in a well ventilated area. if this does not help it is more than likely either due to some sort of heat sensor problem or something to do with the fans.

Nothing is tied to the specific game disc on either the PS3 or Xbox 360 as far as I'm aware. You didn't specify which system. You could start a game on a Redbox rental for instance, and then continue it on a purchased game. As long as you have the save game data still, it should work.