I think you should explain that you're OCD - like Rainman - to everyone, 'cause at this point you're just a jerk.

Don't do it. You can get a lens to put in front of the projector to change the output size. I don't know where to get them, but I have seen them in use on home theaters

That in-wall toilet system most likely has a bidet connection, but you would have to know where it is and drill a hole in the tile to get to it.

I’ve never seen a projector mounted in a ceiling, other than lifts to conceal when not using.

How many inches of throw do you think you will gain by doing this? I can’t imagine much, if any.

And how does this get you the height you need? Most modern projectors can project from just up any position at all, ceiling, wall, table, etc…?

Since your toilet is set into the wall, you'll want to tap into the faucet line instead. Simply get a splitter and add the washlet hose there. The washlet will replace the toilet seat on the toilet.

tl:dr get a narrower projector.

will cost less than what you are thinking of doing, and will not result in your house falling down.

I forgot to add, this is what the toilet looks like https://i.imgur.com/zP4X79A.png hence I was looking at the sink

Totally right this is the way it’s done when framing, make sure you add hangers on the small pieces.

Here is an install video from Tushy showing all the steps. https://youtu.be/jZKxmk_dIVg

Looks like the hinge is bent. Get a new hinge, use 2-3" screws for additional support. If screws won't bite after reinstalling the hinge, throw some toothpicks in there with a bit of wood glue, let it set and screw away.

When mine does this I usually suck out all the water with a pump action plunger, tip a few pints of boiling water into the drain hole then carefully use the plunger on it until it unblocks. With hot water you can feel your drain pipe until you find the spot where it goes cold and that's your blockage point.

I can't speak for all bidets. I got one from Tushy and it came with a short metal hose and T connector.

It would be no problem to hook that up to the water connectors you show there. Step 1) turn off the water supply by turning the cold water knob down in the picture. Step 2) disconnect the water hose and install the T connector where the hose was. Step 3) connect the sink hose and the bidet hose to the t connector. Step 4) install the bidet and connect the other end of the bidet hose to the bidet

At this point it should be all set up with child water going to the bidet. If you are thinking about hooking up the hot water instead you should consider some things first. How long does it take your sink to get hot water flowing when you turn on the hot water only? In my house it can take a minute which is longer than you'd use the bidet, so no point in plumbing hot water. The other thing to consider is how hot is your hot water when it gets flowing? If it's hot enough to burn think twice before pluming that into the bidet. You don't want to burn your ass.

There are kits sold in Amazon or homedepot that have all tools and parts to connect your bidet to your bathroom plumbing. Also clear instructions.

That can easily be done. Sounds like you have zero experience. Call a carpenter. He can sister each joist on either side of the joist that you want cut out. Then he will head the joist off with hangers etc. I don’t know if the prize is worth the price though.

yes, and look at how shitty they are and how often you have to repair them. proper, wedge shaped nails work way better for long term. round nails suck for holding things together where they will expand and contact and be moved around on a regular basis

You would not be attaching it under the sink unless you plan on shitting in the sink. You would utilize an adapter for the toilet tank feed line. It will allow water to fill the tank and to also feed the bidet. Here is an example T valve for bidet

Also check where the dishwasher drains into the pvc sink drain. Mine was recently clogged at that point and cleared it out just but pushing through with a screwdriver.

Nope

There is a chart showing the size of.joist and the size of notch permitted.

4" is definitely not listed there for this floor system.

Those look like 2x10 to me.

Contact a Structural Engineer.....$500 visit could save you thousands in structural damage repairs by compromosing a joist, near a structural wall.....seen it dozens of times over the years....always expensive to fix!!!!

Is there a smaller projector out there?

it is possible if you hire a professional to review the changes to the foor framing and design a new load path, but that is a lot of work just to mount a projector. i'd find a new way to do it.

Fingers crossed for you.

Just buy a ceiling mount that lets it hang down a bit so it's below the joists

I mean— minus the whole driving to a harbor freight part. OP already has a hammer.

My solution is free, and as a tip it’s generalizable, and something a lot of people don’t know— if you blow through trim with an 18 or 15 gauge brad, bending the nail back and forth until it snaps is a lot less destructive and easier than trying to back it out with a set and hammer.

It's prob cheaper and easier to just buy a smaller/different spec projector since doing this right would involve at least 1-2k to have an engineer make sure it works plus another 1k for labor and parts . Easy 2-3k for the whole project and a good projector is only 1-2k.