Theis repair is called a "sister joist".
Ideally, they should extend from end to end a and rest on the support that the original joist sets on.
If the original joist is on a hangar, they should install hangars on eah end and space the new joist, "1 joist away" so they have room to screw...
I'd close all the doors and run heat, fans and a dehumidifier for a week or so to encourage evaporation.
Any loose paint areas should be stripped and spackled smooth.
Wait a few days for that to dry.
Buy a STAIN BLOCK PRIMER. Oil base works best.
Use a latex paint with mildewicide.
Install a HOLDING TANK in the basement in the waterline before the toilet water line. The water will warm up before getting to the toilet and your condensation will stop.
Your shed dormer is probably not vented. CoreVent makes a ridge type top vent where the dormer meets any sidewalls.
You need soffit vents in combination with ridge vents to correct this problem