Can’t get bathroom light cover off!

If the wood is in good condition and you refinish it properly I see no reason not to keep that flooring. Most oil based flooring vanishes will work fine but if you want to get really serious have a professional flooring company come in and finish it with an aluminum oxide based finished or one of the catalyzed varnishes.

Vinegar is not going to be effective at remove the tiles, but what is effective is a bigger tool. Since you are looking to save money, you might be able to get away with using a cheap scraper. It will look like a putty knife but a lot stiffer and have a wider blade. Then using a hammer (wear gloves and glasses), just start tapping the scrapper under the tiles to get larger pieces up.

Once these tiles are up, you will need to get your sub floor (the cheap plywood) to be smooth. If you got nail pops, rough areas from glue or gashes these will need to be leveled before you put the vinyl down. Vinyl is not very forgiving in hiding defects.