The 180 days concerning becoming a tax resident is cumulative during the year. It doesn't reset when you leave and return. Your visa resets, but not your tax liability.
Any qualifying school must be registered with the ministry of education. The Way Pho, and the ITM massage schools are registered, but I don't know anyone that has tried that route, so it just might be accepted without too much hassle.
In most cases you can apply for your wife's DTV as your dependent as soon as your DTV is approved, and in many cases use your bank account, with a sponsorship letter from you to meet the financial requirements for her. Not every embassy allows this, but most do, after you get your DTV.
that is pretty much impossible, as any agent will not have access to the separate room mail-in 90 day reports are processed. It is not the same section A that in person 90 day reports are done, and where agents have access to. Just hope and pray they mail the passports back to you. That is your only option at this point.