What kind of resident do you mean? If are here 180 days or more in any calendar year you are automatically a tax resident but other than that have no particular residence rights
Yes you should remain in Vietnam until the process completes, they can ask for proof you are still in the country or even invite you for an interview. You can't get a sticker in your passport, PDF only, which you print out and show immigration along with your passport
You can't have an adult child dependent on your DTV. It is possible to get an adult child dependent visa but you need to be on a long term non-immigrant visa first. It's also going to need medical certification and agreement from immigration so its a difficult option to get.
I'd strongly advise you to go and get independent advice. It's not about the training it's what you do after you are trained. It might be worth you looking through some of the jobs here and seeing the requirements
times change. You need to have a bachelors degree now to get the Non-B visa that allows you to work inside Thailand as a teacher. Here's a link to the requirements for immigration to issue a visa in Thailand, look at 5.5 and 6.6
So you arrived on a tourist visa, applied for and obtained a 90 day Non-O visa inside Thailand (which will be stamped used). Your visa is now not relevant, all that is relevant is the date you are permitted to stay in Thailand until. You can apply for a 12 month extension of stay (not a visa) when you have 30 days or less left on your permitted to stay stamp, some immigration offices allow you to apply when you have 45 days or less. Best to check with your local immigration office and also get their handout of requirements
do you have a bachelors degree? If you don't its going to be difficult to get a teaching position, TEFL is not enough. What the right visa is depends on your situation. If you want to come and teach you will first need to find a job and get your employer to provide the paperwork for you to apply for a Non-B visa