Just be careful removing the drain so you don't twist the pipes under the tub or break the drain. They come in several different sizes and some have fine thread while others have course threads and make sure you don't loose the rubber gasket that goes between the drain and the tub on the bottom side. When you reinstall the drain make sure you get it good and tight. There is a special tool called a spud wrench for this and I would suggest that you have one. Good luck

I like to use bondo to fill in damaged wood and then just sand it very good and I've had great luck with it. I just recently fixed a hole in a hollow core door from the stopper. It turned out great.

A grinder would work good to grind the unlevel ones flat and then you can lay the new ones on top. I wouldn't use glue to try and make up space because it will probably leak out before drying. I use use mortar if you didn't mind having a large joint under it.

Just throw a wedge in between the concrete wall and the back sides and push them out a little bit.

You should have enough room behind the flange of the dishwasher to use a piece of 1/2" all thread rod and a couple of wall plates which is what would mount to the floor and top ( may have to alter the plates some to keep them from sticking out. I would also check and see if that style of dishwasher would allow you to push a piece of 1/2" plywood up against it and then screw it to the top and floor.

Be extra careful so you don't sand through the veneer it's easy to do.

On a tank type toilet they use the atmospheric pressure from the water in the tank to create the flushing action but a toilet without the tank needs the volume and pressure from a larger pipe to do the same thing. A toilet flush valve is always 1" and sometimes if the pressure is to low it still can have problems flushing. Also the flushvalve needs a copper pipe coming out of the wall that is supported so it cannot move so the fitting to connect it can be soldered on and you don't want it to be able to move around or that will cause leaks where the pipe goes into the bowl.

There is no reason you can't do that but you will need a 1" water supply to flush it. The flange would be easy enough to change either with a repair type flange or anchor bolts going into the floor where the mounting bolts come thru the bowl. One thing to keep in mind is that those types of toilets flush quite loudly so it could a issue for that.

If your cabinet is secure and the top is what is moving some when cleaning I would suggest using silicone under the countertop ontop of the cabinet. There are generally corner braces on the cabinet where you can put the silicone thick enough and then let it dry well. After that you can caulk the joint with whatever caulk you like for the look. This is the method I used as a plumber installing bathroom cabinets with non wood tops. Good luck