If the landlord registered you electronically, you should get a receipt or screenshot of the registration. You could do it yourself at the immigration if you bring your lease contract and a copy of the landlords housebook and ID-card. It’s the landlords responsibility but your problem if it’s not done.
Yes it should be enough if you also have a receipt of your TM.30 address registration at your immigration. The owner of house should register you on a TM.30 within 24 hours of your arrival.
If you don’t have any recent excessive travel history on visa exemptions, you should be fine with a re-entry and still be considered as a genuine tourist to Thailand.
You should be able to a apply for a Non O in country on a TM.87. It’s known that some smaller immigrations don’t offer this service. Where du you stay?