Excellent discussion and advice so far in this thread. However, there is another possibility for water access that hasn't yet been mentioned. It's hard to see details in the one picture that shows it, but it appears that the roof covering the space between the house and garage was improperly attached to the house. It looks as though the roof structure was simply attached *through* the existing siding, so the top edge of the flashing at that joint is on the *outside* of the siding. There are gaps at every joint in the siding, and it's impossible to see how (or even if) those gaps were sealed. If water were able to get through between the siding and the flashing, run down the wall to the ceiling, which is right about the top of the T-111, it could enter the *top* of the wall, run down and collect at the bottom where the mold is forming. You should make sure the joint between the top edge of the flashing and the siding is well sealed (the top edge of the flashing should really be *under* the siding, but to fix that would mean taking the entire roof over the walkway down and rebuilding it properly). I would also take down at least a small strip of the ceiling so water could escape by running down the outside of the wall instead of finding an easier route *inside* the wall. Hopefully the main structure of the house has not been compromised and there's no rot there. Probably should check just to make sure. Personally, I would remove all of the ceiling material. It conceals a large void, inhibiting ventilation, but also creating an unseen path for fire between the garage and the house. Even if it was noticed, it takes a few minutes to tear down enough of the ceiling to get water throughout the the entire void, and that could be a couple of minutes too many. If you want to keep it, I would advise cutting a scuttle hole near the center, fairly easy for humans to open (but not for raccoons, etc.), easy to get a ladder through, and big enough for a large firefighter in full gear including breathing apparatus to turn around in. It comes in handy if you ever want to run wires, etc., through there, too.
Sorry to run on so long, hope it's been some help. (Disclaimer: I have never been employed in the construction trades. In fact, with 20+ years in the fire service, I've done a lot more un-construction (or would it be de-construction, or maybe just destruction?) than construction.)