I'm Lucy, and I'd be happy to answer your questions today. I'm sorry to hear about your situation.
1. If you serve a 3 day notice to pay or quit, and he actually pays you, you have to let him stay. But you could turn around and serve him with a 7 day notice the next day, so it's really up to you. If you want him out ASAP and don't think he'll pay $7,000 to stay, it may be to your benefit to serve the 3 day notice.
2. After he moves out, you can sue for the unpaid rent and get a judgment against him. If he has a job, you'll be able to use that judgment to garnish his wages until he's paid in full. You can also report the judgment to the credit bureaus, which might encourage him to pay faster if he's worried about his credit rating. A judgment can also be used to be a lien on money in a person's bank accounts.
3. You're allowed to serve a notice of abandonment at the same time as a 3 day notice. If you serve a 3 day notice and he doesn't leave, then you can go to court and get an eviction order (plus a judgment). If he still doesn't leave, a sheriff will come to your house and place his belongings in the street. If you serve a notice of abandonment, he has 18 days to reclaim his stuff. Given the practicalities of the court system, chances are that the 18th day to reclaim his items would occur around the same time as a hearing on the eviction (assuming he doesn't pay). So it's up to you if you want to try it. Note, though, that if you go with an abandonment notice and he doesn't reclaim his stuff, you'll have to hold an auction if it's worth more than $300. This site has some good information on that.
4. The phrase "premises in possession" is fine even if he only has part of the property. You can add other language if you want to try to clarify it, though.
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