Do I have to hole out on every hole, and how does Stableford scoring work?

Medal (stroke play): yes. Under Rule 3.3c you must hole out on every hole. If you don't, you must correct it before making a stroke to start another hole (or, on the last hole, before returning your card), or you're disqualified.

Stableford (Rule 21.1): you don't have to hole out. Points are based on your score against a fixed target, which is par unless the Committee sets something else. In a handicap competition your net score is compared with par.

• Once you can't score a point, you're encouraged to pick up to keep play moving.
• If you hole out and score points, the card must show your actual score. For a zero-point hole, you can leave it blank or write any score that gives zero points.
• A few breaches that would disqualify you in a medal (such as not holing out) only cost you the points for that hole in Stableford. Other disqualification penalties still apply.

Maximum Score (Rule 21.2) works the same way: if you pick up, you get the Committee's maximum score for that hole.