You can file a TM30 yourself at Chiang Mai Immigration - need a copy of his tabien baan, lease/rental agreement and a form. If the address is already on file, only the rental agreement and form. Source - I've done both ways before.
You can't apply for a tourist visa within Thailand - you need to go to a neighboring country or back to China to apply through an embassy or consulate. There's also no three-month visa for a tourist - different if applying for a retirement or other type of visa. There is a 60-day single entry tourist visa that can be extended for 30 days at Immigration.
With only one previous visa, it shouldn't be a problem. There are no reports of people being turned away by Canberra or Sydney for applying for additional visas.
Never had a problem using my Bangkok Bank and SCB cards before - ATMs and tap'n pay, but I usually try to use cash - although that is getting harder in Australia.