European Economic Area to be precise (i.e. Norway, Switzerland etc abide by the same rules).
Yes, if you're not a tiny business (OP isn't tiny) you have to pay VAT to each end customer's country. Though you can declare it in your home country to reduce bureaucracy.
VAT law is well developed, what matters is the location of the end customer, not where the seller is, so your technically speaking isn't how it works.
Wrong, if they buy from US Amazon, they pay VAT at EU customs entry (unless customs misses it or it's below threshold of some 40 EUR or so).
Is it enforced? Yes. Do they catch everyone who cheats? No. Do you want to break the law? Better not.