Thanks for the group suggestion. I've posted over there as well, but no response so far.

No to garbage disposal, the flap separates the basin from the pump, the hoses don't have built in backflow stoppers. I'm about to hook the pump back on to the basin and test it without the takeaway hose, try to further isolate the problem.

I’d look for alternatives, modifying the house support structure isn’t something to take lightly. For example, can you hang the projector just below the floor joist?

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JFC don’t do that lol

Definitely doable, but I’m not going to give you structural advice…. Double up both joist on both sides cut out the middle add some hangers and a new beam to support the middle joist….

Ask the folks who work on them for a living: /r/ApplianceRepair.

Yeah - this isn’t a “wing it” kind of DIY job.

Nope, that's the entire end of the joist, which is supported by that wall.

You could probably notch an opening in the wall and mount projected on opposite side of wall and light goes through opening.

Yes, grab a drill (or better yet, an impact) and suck those screws in tight. I replaced all my door's upper hinge's middle screw with a 3" because they're just heavier doors and needed the support

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Given the screen in the wall in the bottom half of the picture I’m guessing it’s a short-throw projector, which get mounted directly above the screen and project mostly down rather than the more traditional kind that get mounted away from the screen and project forward towards the screen.

They’re just trying to avoid moving the top of the screen down to provide space for the projector.

If you can grab the head with a pair of vice grips, you might be able to bend them back and forth until they break off.

A $20 Harbor Freight angle grinder with $5 in cutting wheels will do the job 3x as fast and with 1/2 the effort.

https://structuretech.com/notching-boring-dimensional-lumber-joists/

Tldr, if you’re even able to (unlikely) it can only be up to a 4” notch.

In that case, you probably want to get some oil in the bearings so they will last longer. 50 year old oil isn't trustworthy.

Can't quite tell the source of the sag in the pics. In my experience it is usually from the hinge pulling loose from the frame or weight pulling the frame away from the opening itself. I usually replace one screw on that side of the hinge with a 2"-3" screw that reaches all the way to the wall framing.

If the hinge is bending though it needs to be replaced.

Nope

Besides the cord, it is in excellent operational condition.

No

Ive always wanted a Universal Hex/Allen Wrench sizing chart with slots to put in the key to confirm lol... so many of them.

It's probably 20 bucks at harbor freight and comes with a grinding disc. Actually I just checked and they sell one for $15 lol

If the blue represents the new supports & the black is what you are cutting out, you have an issue. The supports won't support what you are cutting out. It MIGHT be doable if you can add joist parallel to the cut one but any time you adjust the actual support structure, you need some professional advice from an engineering direction.

I'm also confused by how placing a projector that high will angle the image correctly. Yes, I know they have electronic keystone correction.

No.

Your floor joist is sitting atop your wall, which is supporting the weight of your floor above. Adding the supports shown in blue will not provide much support to the joist where you cut it. You risk having some serious sag in the floor above, or structural failure at worst.

Unfortunately, I don't think there is much you can do in this instance.

I wouldn’t.

https://buildingadvisor.com/notching-and-boring-joists/