We if you leave without a re-entry permit you won't have a valid extension anymore.
I hope you are actually learning Thai too because on re-entry you are going to get questioned in Thai and only Thai and if you cannot answer they will deny your entry.
Assuming the kids are still young, they just go on the back of your wife's DTV as dependants (your wife submits an application for the DTV, once accepted you can apply for dependant DTV visa, if they will be going to school here you also have the option of Education Visas for them which the school would sort out for you.
depends on the office, and even the IO you see. Some will want it some won't. Better to have it with you, just ask for a copy or for a file to be sent to you on Line or another messaging service.
TM.30 is filed by the hotel, agent or landlord of where you're staying. If you're renting a condo or Airbnb talk to the owner first, even if you've already booked and they won't file one for whatever reason, you can get aroind this by staying in a hotel for a couple of night and get them to do it.
Not unless your embassy in thailand does income affidavits. If they don't (which most don't these days) you will need the 800k for the first extension.
That is your 90 days notification date. 90 days is called a notification of residence. You get 15 days before and 7 days after to report. After those 7 days you are fined 2k.
As for what documents you need like everything, it widely varies office to office, so you should check with your local office.