The "self destruct if I am not protecting anything and nothing is assigned to protect me" only works when the planet is no longer owned by the AI*
As for your questions.
1. It is supposed to check on death of a ship or a ship leaving the forcefield because it was freed.
2. Yes, buildings and engineers count, except for other forcefields IIRC (so if it has a alarm post under it, it will stay alive. If it has another forcefield under it and nothing else, and the AI does not own the planet, then both the post and the force field will self destruct)
3. No, forcefields provide protection against tractor beams.
*Or more accurately, its actually if there is no reinforce warpgates on the planet, but in most cases, that is the same as saying when the AI owns the planet, as the command station is a reinforce warpgate.