it's best to get it from the office to fill it out. Some offices are very particular about their forms and if you print your own it may be rejected for being double sided, or not being doubled sided, or the wrong size paper, etc
yes, all visa exempt unless you had a visa. There used to be a rule about 2 times per year over land without a visa but that rule is gone now when they changed visa exempt to 60 days.
you didn't get a tourist visa and nothing has changed about the tourist visa. You got visa exempt which is not a visa at all. It's in the name, exempt from needing a visa. A visa is something you must apply for and pay for.
I'm going to ignore that most of what you said at the beginning is incorrect.
You can go to the immigration office that matches the province where you are staying and apply for one 30-day extension. Most offices will accept when you have 30 days or less left, though Chiang Mai will allow 45 days or less. They are strict on requiring you have a TM30 from where you are staying I'm Chiang Mai. I believe Chiang Mai has 2 immigration offices you can use for this, one near the airport and the other at one of the malls.
Probably easier for the country which you don't have a passport for. Because if it's your home country then the assumption is you are trying to get her there to bypass the immigration process and marry her and then apply for a change of status. But if you're not a citizen of the country you want to take her to, that's not a risk.