If the building is original and hasn't been renovated, then the studs could be anywhere from 12" centers all the way to 24" on interior walls due to the age of the building. I've seen some with 4 foot centers as well.

You shouldn't drill through the door to run an electrical cable, that's just an accident waiting to happen. Where are the panels and batteries located exactly?I'm guessing the panels are in your yard and not on your roof?

The best advice I can give you is to sister some new temporary joists to the existing. They'll either be 2x10 or 2x12 that you'll need. Do not remove anything existing either,as you could easily compromise the structural integrity that's there now.You'll have to pull down most of the drywall on the ceiling to accommodate the sistered joists,as you'll have to join the temporary joists as far back onto the existing where there is no water damage with an overlap of at least 6 to 8 feet on both ends,or worse case scenario, all the way to your top plates on the outer walls. If the joists are bad enough, you may need to sister on both sides of the existing with cripples in between the joists for more structural integrity until you can properly fix the problem

It could honestly be a number of different reasons, not enough ventilation on the roof, foundation, climate, age of the house, windows, doors,etc.I would get someone to come and take a look for a proper idea of what is going on

That's one thing that has never changed in the 30 years that I have been doing contracting.Accessing taps have always been a huge pain in the ass lol.I just quit messing around from underneath and opt to pull the sinks off the vanities instead, unless it's an outdated home and they still have copper.

Caulk around the entire exterior of the window frame, fill in any gaps between the window and siding.If you have casement/awning windows run a bead around where the window seats into the frame. Sliders same thing, run a bead around the outside of the window and track. In the case of the mold,you will have to expose more around the window because you don't know where it is leaking from and how much mold there actually is. It could be halfway up either side of the window or even started from the top for that matter.Vapor barrier and all affected insulation needs to be removed.After removing all the affected material, you then need to saturate the framing with a solution of bleach and water about 60/40,then let sit for 8 to 10 hours depending on how much mold there was.