Can I exit and re-enter Thailand on a student visa, extend it for years, and apply for a Schengen visa from there?

If you have a visa/extension you can apply for a re-entry permit, which will allow you to leave and re enter. You’d be stamped in to the same date that you previously had.

You can only extend a student visa for a max of one year, and even that may not be possible. Some courses/schools will be 3 months per time.

You can apply for a Schengen visa in Thailand. Usually you’d need to contact the embassy of the country you’d be going to be in first, which in your case would be the Spanish embassy.

1) re-entry permit

2) if at university yes, if doing non-university course then it's much harder and unlikely

3) yes but you need to be attending class in Thailand or your student visa will be cancelled by the institution. Unless you have a dodgy visa of course...