When someone joins, they get their own private link — that's the only place their assignment ever shows up. There's no public list of who has who; each person can only ever see their own match.

Everyone is randomly matched up so that nobody is ever assigned to buy for themselves — and if you use the "avoid repeating last year's pairs" option, nobody gets the same match they had in that previous event either.

Yes — anyone who's joined can be removed from your dashboard right up until you draw names. Once names are drawn, the participant list is locked.

At least 3 — with only 1 or 2 people there's no meaningful way to keep a gift exchange secret.

Yes — just create a new Secret Santa event whenever you're ready to run another one. Each event and its join link are completely independent. When you draw names for the new one, you can optionally enter last year's event code to make sure nobody gets the same person twice in a row — matching is done by the name people entered when they joined, so it works best if people use the same name both years.