I'm wonder whether, as an American on a visitor's visa to Greece for a month or so, I can work for English company on my computer, and if so, if there are any rules/legal restrictions/taxes etc. I should be aware of. I'd appreciate any advice!
For the first three months, you have no restrictions; you are covered by your Schengen visa (i.e., your American passport, valid for at least 3 months from the date you LEAVE Greece). If you want to stay longer than three months, BEFORE leaving the US, you have to start your residence permit application; it's very involved but I've been through it so if you have questions, feel free to ask. For us Americans it's an entirely different procedure than for the EU citizens who make up the majority of this board. A lot of people will tell you it can't be done, but that's not true - it's just very intricate.
But for the first three months, you don't have to worry about anything. Just do your thing, no one is going to care what you do on your own computer.