You have to apply using the consulate that serves your current location so if you're in Ethiopia you can only use the consulate in Nairobi. How it works is that you apply for your DTV and once it is granted your wife and kid then each apply quoting your DTV approval and providing proof of marriage and parenthood. They also apply using the consulate that serves their location. Note that you can't apply via any of the US consulates if you are physically in Ethiopia
I'm guessing you need to leave the country because you either don't have a bank account or don't have long enough left on your stamp as I'm pretty sure Hua Hin can do in country visas. You can just leave to any nearby country and apply for a 90 day Non-O visa based on your age. Your wife would need to apply for a 90 day Non-O visa as your dependant, however many consulates will not issue this based on you being on a visa based on your age, you can try but don't be surprised to be refused. Some consulates will issue it based on you being on a Non-OA visa but you can only get this in a country where you have an official right to reside such as your home country.
yes they are but I'm not aware of immigration offices inside Thailand accepting same sex marriage for the purposes of obtaining an extension of stay. I would check with your local immigration office where you will live in Thailand and that you will apply for the extension of stay at.