This is the first time I hear about a birth certificate to marry a Thai national. Usually as a first step we are going to our Embassies to obtain an Affirmation to Marry certificate and that usually just requires a passport.
Yes, you can if you have what it takes (bank account with 800k thb and so on), not at airport tho. You must visit your local immigration office to apply.
sorry to burst your bubble but I actually spend far more than 30 dollars a day. I own business too and it happens to be quite successful. I stay in Thailand because I can. Does this hurt your boot? Your problem.