I travelled around the world for 6 years and the number one impediment to staying in any country was the visa situation. If you are staying there longer than a tourist visa, and especially if you want any kind of employment (or are in a situation where you are visibly employed) the red tape in the vast majority of countries is very tough.
The only exceptions I know about are Chile (which gives one year visas that are relatively easy to renew) and Panama (which will give you residency if you start a business there, it can be a small one.)
Poland does not require visas for US citizens (even though US requires visas for polish citizens).
As for working you could just incorporate. It costs maybe 30$ and most hacker-type polish friends can help you with that. Later it costs $250 per month to own a company (mostly social security fees and health insurance, not sure if this is obligatory for foreigners but probably yes, plus service to help you deal with paperwork).
No, US citizens can enter Poland (or more correctly the Schengen area), and stay there for 90 days in any 180 day period. This does not give you the right to work there.
Chile gives "rentista" visa TE6 if you can show some sort of passive income starting with ~$800/month per person. This visa can be converted into a residency in 1 year.
The only exceptions I know about are Chile (which gives one year visas that are relatively easy to renew) and Panama (which will give you residency if you start a business there, it can be a small one.)