How old is your son? Mine has lived here all his life on an O visa extension as my dependent and has never done a 90 day report (he's nearly 8). It's never been mentioned or been an issue when getting a new extension. I do mine and my wife's online.
An IDP is just a translation of your national licence. I don't believe it includes a Thai translation! That said, the police don't care and are looking for a reason to fine people.
Australians can't buy a visa on arrival but they can enter Thailand free of charge 'visa exempt'. This currently gives them 45 days and can easily be extended by another 30 days for 1,900 baht. Alternatively they can buy a visa from a Thai Embassy/Consulate in Australia which gives them 60 days, also extendable.
The Ed visa extension is finished now, so I don't think he has any choice but for the school to restart the process again. I wouldn't overstay, so he'll probably have to return to Taiwan and apply for a new Ed visa there. Unless the school can somehow do this in-country.
The TM30 needs to be filed within 24-hours of moving to your new address. The 90-days isn't reset; you just report as usual but you'll have to do it in person at your new Immigration office, where the TM30 was filed, the first time.