Legally the landlord can't let anyone into your apartment, not listed as an occupant, except himself or one of his agents (assistant, repair/maintenance worker) and only with proper notice as required by state law, except in cases of emergency.

1) it depends on the lease

2) the landlord can certainly let in repair people

3) other than that, i don't think so

You are not clear. He can accompany a prospective tenant to see the place. He can authorize a tradesman or rental agent to go in.

Generally no, without your consent. Unless it is stipulated on the contract of lease that you signed. Re-read your contract and take it from there.

Not "anyone." Only those necessary to perform repairs or maintenance.

He can't come in without notifying you ahead of time.

i was in your apartment last week. Nice!

No only himself