How does immigration law work for this kind of trip? Do you need a work permit/visa?
Always, always, always, always, always check visa requirements. If you need one and don't have one, you will be buying a very expensive plane ticket when you get turned back at the border. Default tourist visas for American citizens is 3-6 months most places, but you always have to check. Don't trust any website that's not the official government, and call an embassy if you're at all unsure.
We are chattel.
In practice just pay taxes in your country of "residence" not the country you're in temporarily. Many countries have 60-90 day visas for visitors from first world countries.
I've received many different explanations over what's acceptable and, of course, many government documents will seem to contradict each other. However it is not beyond the possibility that one would run into a problem with this when dealing with some local authorities.
The other thing to keep in my is that residency is often tied to a minimum number of days spent within the country you are filing taxing in. They also can take into regard your "intent" of where you plan to be living next year. You can be judged to not actually fulfilling the residency requirements and be taxed on foreign earnings.
Most people who do this won't have issues but I've witnessed had one border agent mull over whether to deny entry or let them through, based on answers to questions about activities in other countries.
Disclaimer: this is not legal advice