Look up mold remediation with borax. Spray the wood with that before you button it up. Since it is an interior closet wall, I 'd replace the drywall. It does not have to look pretty and it will be good practice for some other project that might be forced on you. Black spots might or might not be mold. Wood flooring can turn black from water exposure only. I think I would let sleeping dogs lie in that case.

I had to do some temporary fixes with windows that were rotted that looks similar to yours. I just had to wait until I could afford it. I used that wood hardener like you did. To fill in the void after digging out all the rot I used whatever, small pieces of wood. I would put some wood, or something solid under window to keep it supported. I also used some spray foam, use the stuff for windows and doors, it is not supposed to expand so much. I also used some wood putty (not the stuff used for filling nail holes) There is a wood putty that is ready to use, I think made by the same place that makes wood hardener. The other wood putty, it's a powder that you mix with water, Durhams. I then painted/primed to give it more protection...I had 5 windows that looked similar to yours. When we got enough money together to replace we had to replace the bottom framing and a stud or two on a few of them. Yes, I do think the sill/bottom is a support. They are nailed into the side, but the bottom I am sure is "what holds it up." Also, I spray some bleach around the area to kill mold spores, I read somewhere a mixture using borax is good for that too. But after wetting it with the bleach you need to dry it out really good before you seal it all up. The two windows that I had to have studs/bottom 2x4 (that supports the window) replaced I had to tear out drywall and dried it out using fans and portable heaters.

I once rented an apartment in Northern California, near the coast. It had a pair of french doors with the glass held in place with tight moldings, but still a driving rain coming off the ocean could find its way right through, water running down the inside of the glass. It was amazing to watch. This obviously isn't what you want to hear, but... perhaps a 1' overhang isn't enough? Your door should be tightly assembled, but it shouldn't need to be a submarine hatch. BTW, I wouldn't glue those panels. The whole reason for frame and panel construction is that those solid wood panels get wider in damp weather and narrower in dry weather. If you glue it, the next time there's movement only 2 things can happen: the glue breaks, or the panel does. Of course the finish will slow vapor transmission and reduce movement, but it won't eliminate it. If you must, a flexible caulk is a better choice.

The problem you are describing is often the result of improper flashing around the outside perimeter of a window. Typically one might need to pull off or cut into some of the siding around a window to reflash it. However, in your situation, since it's a brick home, it could be more complicated if the brick is tight against the window and there is no way to get to the flashing without brick demo, which I know you don't want to do. Here is a diagram that may help you understand flashing with brick veneer: My first call would be to a window installation company to see what's possible. Maybe caulking alone could resolve this, but it needs a proper assessment. As millworkman notes, posting some pictures here (inside and out) would help.