A humble sailor
The only quick fix here is getting a good floor fan and put it at the top of the stairs. If i understand correctly using a fan for circulating air should be done by pulling the air (ie setting it at the top of the stairs) rather than close to the mini split and blowing the air where you want it. Running the exhaust fans should also help to do this exact thing by pulling the hot air out of the ceiling areas and allowing the cooled air from other rooms to enter the bedrooms
No. The split unit won’t have any way to move the air through the duct or attach to the duct. And the duct is undoubtedly not designed for cold air to be moving through hot spaces which will cause condensation on the duct/ and mold in the spaces where it is run.
Lol. No. Gonna get syphoned out ever time a toilet flushes past that or every time it pumps. No air gap/ directly connected prohibited by codes in most places. I mean it’ll work until it doesn’t. The proper way to do this is with a laundry tub dumped into a proper vented container/basin pump. Or run a drain over from that other pit and put a 48" stand pipe with a 2" trap.
And the more you open up to fix the more systemic problems you find. In an old/fixer house. Try to pick one project at a time. Spend some money when u have too. Have seen some smart people locate home improvement grants for upgrades/HVAC. You’ll need to spend money. No avoiding it. In a couple years you’ll forget how bad it all seemed and get used to the minor annoyances you haven’t gotten too. And don’t forget to learn some things. Buy a bug pump sprayer and specific chemicals on line. Learn how important gutters are and getting storm water away from the foundations. Get some rugs to cover the gaps and caulk the cracks. Most importantly A good partner for the decorating to make u forget it’s a shitty house no matter how much work you put in
To answer your question I just drink a lot.
Left ones got a Netanyahu vibe