Ia it safe to run a 70w HPS bucket light inside? Will it like, catch anything on fire? Carpet? Also, will it overload the breaker?

Catch fire: Depend on quality. If you want to be sure, add a ballast thermal fuse and/or add a slow acting breaker to 1.5 * mains current (written on the ballast or fixture label and/or in the manual).
Trip a breaker: If everything is working OK, don't worry, the breaker is there to act, if something really goes bad. If it does sometimes trip, ballast is dying and you should change it. The risk in relying inly to this breaker is, the 15A is too much to reliably protect the ballast, it will serve as a last-stage protection, if everything else fail.

The only concern I have with NEMA bucket lights indoors is the possible overheating of the ballast if air circulation is poor. I have a couple in my workroom and use fans on them because they are close to the ceiling where heat builds up and the fixtures get extremely hot.