Not everyone has the resources to make a working place anywhere viable. You seem to have the resources to make a hotel in Budapest or whatever work. Talking so causally about international travel already puts you beyond most Americans’ income.
So yeah it is very, very easy for folks with a job like yours, and with the resources you’re afforded, to work anywhere. The issue isn’t proximity to resources, but the ability to complete work.
Usually true. Although some countries (like China) do have a business visa that you're supposed to get if you're e.g. attending a business event.
With a US passport traveling to countries for example, there's nothing keeping you from visiting for a month or two and working from a short-term rental. I don't know all the theoretical rules that may exist but I know traveling partly/fully on business, people obviously do work for their US company all the time without filing any special paperwork.
https://well.blogs.nytimes.com/2016/05/02/a-shocking-way-rea...
https://www.google.com/amp/s/www.bloomberg.com/amp/news/arti...