Not happening, I had the same problem with HMRS tax refund from UK being a cheque and eventually I had to give up.

No bank in Netherlands would honour any kinda cheque so I still have 1k Brit pounds that I can’t access.

Some EU countries have a 180 day visa free regimes (For US passport holders), and only if you stay within them, though if they check is a different story.

Only if you enter via major ports of entry.

Now if OP were to leave via ferry or a land border from Greece to Turkey then its a different story…

Tell the person who stamps your passport that you’re visiting their country with the intention of working remotely for a foreign company and see what happens.

I know you taking the piss now, but in Canada thats explicitly allowed on a tourist visa.

In the US though, yeah you getting denied entry haha.