Calls for the Hurricane Katrina act.
Plastic/tarp from peak to below skylight.

Don't cut the beam. Hang off both beams with your hangers (perpendicular). Then install you other 2X parallel with the existing joist and directly under your seam.

The best duct sealer I have used is like a paste. Easy to force into holes/seams and duct does not need to be super clean to adhere.

I would go with three joist with hangers.
Good support and never need to worry.
I would worry about how the water got in and what is growing in the wall.

I have used the gray duct sealer and works great. Wear surgical gloves.
Regarding mice droppings. Check your flex duct if you have a cross over. Replace with metal and pull your old 12 flex over metal so that it is insulated to R-8 and has the vapor barrier.
Also make sure it is off the ground.

See if your local utility has a energy audit program. They may come out and test your duct for leakage for free.

If possible boil about two gallons of water and the immediately dump in kitchen drain. The stoppage is past the traps and in the main line. The boiling hot water will help clear some of the build up in the line. Garbage disposals are a big contributor to stoppages at kitchen sink drain lines. Plus y...

Blow in as much as you can but maintain,1 inch air space below roofing so roof can breathe.
If you re do the roof in the spring you could put on a TPO roof (rubber membrane) and never need to worry again.
Can be installed directly on existing metal.