some photos:

Yes, a carpenter could take care of that. Or a handyman.

It's IMPERATIVE to find the source of the moisture that's causing the rot. This isn't a bandaid fix of something simple. It requires a lot of investigation and a good knowledge of construction to be able to figure out how big of a problem this really is. It can vary from a Problem, to a REALLY BIG problem.

l could not have said it better than lwo.

You need to establish if it is "new moisture" damage, or old moisture damage, though. Since it is such an old house, it could be possible it is old, old water damage/rot from an old leaky roof. Was the stuff you had to clean up damp/wet? Or dry?

If a previous didn't repair that after a roof leak, then this home probably has a whoe host of horrors behind the walls. Better be prepared for a money pit if that's the case.

third vote for finding moisture source & fixing that before replacing stud. let it dry out before closing the wall up after issue is solved &
replacement of stud (and bottom plate...if it doesn't dry).

look up for the moisture source as water travels down.

best of luck.