How would an individual go about doing this? Would they need a good degree?
I have a junk degree and currently work as a dev after teaching myself programming.
I’d recommend a combo of teaching English and freelance remote dev. Definitely possible to make it work..
This depends on location though. Some places you can pick up local jobs in English because there are enough companies that work in English.
As an American you won't get a visa (to the EU) with a story of 'I'm going to support myself by looking for a teaching job and doing some odesk'.
It depends where you are going.
From my experience any degree is ok. And some didnt need that although it did help a little bit with a special visa I was able to get.
I do not know what immigration laws would apply. And I do not know what stack (and on what level) you program in. But at least my employer is actively looking for devs (Germany, Berlin, Hamburg, Munich or Frankfurt).