sanscript: If you are going to bother with it, then you just have try and see the result. Worst case you'll learn something about it. :)
1. Install the game from the home computer. Try also with alternatives, like OpenMW and others. You can't do anything anyway if the schools computer doesn't have the necessary libraries for the game (you can actually ask the administrators if they could help you install something - and many schools should already have a certain amount of libraries in their ecosystem to support various software).
2. Speed. Old games can do just fine with USB2 or even 1, however, waiting for a big/newer game to load when not using USB3 is a pain. SSD disk would be the best option.
3. Some games do run just fine even if it doesn't detect any registry entry for it, so it would revert to the difficult settings for saves, which usually is in the users profile, somewhere. Find out where on your computer and then check for the same folders/files on the schools computer.
4. If you have a laptop/tablet with Linux/Win (or even Android on) why not just go with OpenMW, and take it with you to the school?
https://blog.en.uptodown.com/morrowind-android-openmw/
5. EDIT: Depending on how locked your schools computers are, best option is to install both windows and the game on a SSD disk, with a USB3/ESATA port on both computers, and just boot up with the external disk. But it's also the one option with most hazzle in the beginning to get it to work.
Doing things portable is fun in the beginning, especially with smaller/older software.
Then the hair-ripping and crying starts... :P
1: Done.
2: I've checked and speed is not an issue.
3: I can copypasta the saves, but I'd rather not.
4: Not an option.
5: Not an option. They've locked the BIOS and set the boot order. I couldn't get it to do that if I wanted to.