Remember that a bunch of very thin coats is better than one big fat coat Use the flex in the widest knife you can to keep it thin. It takes practice, it's not an easy skill until you learn it.

I did something like this once for a garage beer fridge using custom cut plywood. I realize it's not the most sanitary idea but hear me out. I got good quality board dense board. Then I painted it with 1 part epoxy to make it as moisture resistant as possible. there was really only cans of beer & pops in there, so I wasn't concerned much. I never had any mold or mildew because I just made sure it was clean and dry. It's a cheap option and you can cut it exact if you have some wood working skills. This is reddit so I would expect a million down votes about how dangerous and irresponsible this is but blah blah blah.

Do you have a dedicated shutoff valve inside the house for this spigot? If so, this is a good project to try yourself without harming too much. It looks like you must live somewhere with little frost because this exposed pipe setup would burst up north. Is that a copper pipe painted white? If it is you will probably have to cut it and put a new fitting on there. Ideally you make the cut inside the house, then mount the new spigot flush to the house and make the new connection inside. Either by sweating copper together or some other sort of pressure fitting.

You would be amazed at how effective just wiping down the wall with some light cleaner and warm water is. If you really feel you need to sand it, get some fine grit drywall sandpaper or screens and a pole sander. Then just lightly go over it to smooth it out. Make sure you get any dust off the wall before you paint it.

I just installed two Mansfield toilets in my house. I highly recommend them, if you decide to get a new one. I got it at Lowe's.

You will have to rotate the flange so the bolts line up how you want. That either means a new flange or a new repair ring that will fit the current flange. You may also have to use a longer water line since it probably won't be lined up straight anymore.

The other comments about plumbing code are valid, but if it's your house you can pretty much do what you want. Just be prepared to put it back if you sell the house, or possibly take some flack regarding the inspections and other potential hassles. It's also possible that if there is a flood/leak or something and your insurance adjuster deems that it was "installed wrong" you could be looking at denial of coverage I guess.