, You need to do a 90-day report if you stay longer than 90 days continuously. You also need to check what documents are required at your local office. Generally, you bring your passport, a filled TM47 form, and a TM30 receipt. Some offices require copies of them or some more papers. Some require only your passport.
, correct. It wouldn't be an issue for now. But if you have extra time during your stay in Thailand, it's not a bad idea to get the documents from your previous school to cancel the old extension. I guess it's another story though.
It wouldn't be an issue for a short holiday. You wouldn't have a problem with entering Thailand again on tourist status just because of not having cancelled the previous extension. It could affect when you apply for a long-term extension, or even for a 30-day tourism extension at a few immigration offices.
You might be better off having proof of being a genuine tourist when you enter Thailand, such as proof of onward travel within given days, accommodation, etc.
, right, once she leaves and re-enters Thailand, then she is no longer "currently" overstaying. She can apply for an in-country initial visa with the correct documents as long as she has 15 "working" days (usually 21 calendar days) at the Chiang Mai office.
, as long as your daughter's nationality is eligible for visa-exempt, one day trip to re-enter Thailand on visa-exempt is not a problem. Then she will be able to apply for an in-country visa.
But It's you and the employer's decision where you want to apply for her initial Non-O visa, because required paperwork at consulate and immigration office are different.