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 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).

==> https://sinnerschrader.jobs/en/

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.

Start off looking for a job. They could help you work through the visa issue, and most people need a job to live somewhere.

I’d recommend a combo of teaching English and freelance remote dev. Definitely possible to make it work..

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'.

This depends on location though. Some places you can pick up local jobs in English because there are enough companies that work in English.