Trimming out sound before or after the start of a vocal is pretty easy. Removing noise from a part of the file where the vocal IS singing, that's a lot harder. There are noise reduction plugins but Ive rately found them transparent for steadty state hiss, and they're not very good for one off things - a door closing, footsteps, that kind of thing. For discrete noise like this, I've had some success with spectral editing in Reaper - a guitarist breathing audibly while playing a quiet acoustic part, for example, and it works maybe a ittle better because you're only modifying very narrow windows at a time - but here too there are limits.
If at all possible, just focusing as much as you can on noise reduction up front, before recording, and capturing as little noise as possible on the way end, really helps. What you can do with technology really is impressive... but the best results always seem like they come from just not capturing noise in the first place.