> I let Time Machine starts backups automatically whenever it pleases. Does it have a hook to run a command to take a snapshot?

I don't know, but you can trigger backups on demand using tmutil(8), and that's what I do.

> And if you do say 3 backups then get rid of #3, can it take incremental #4 on top of #2, or will it complain that the #3 state it expects is absent?

You can delete arbitrary backup(s). "Classic" Time Machine backups are snapshots deduped using hard links. HFS+ even supports directory hard links.