To put it bluntly, you're right. We have to do more work just to get started accepting money than our American counterparts. Yes, the insanely complex EU-VAT rules actually need to be followed. Yes, a lot of your customers will want to pay via bank transfer and not through credit card. Yes, the American folks will still want to pay in USD and not in EUR.
We use ChargeBee (https://www.chargebee.com/) to abstract away most of the pain. I would greatly suggest using such a service (Recurly and Chargify are more enterprisey alternatives). We had a selfmade implementation before switching to ChargeBee, and one EU rule change caused us to have to re-issue all customer invoices for one quarter. Not fun. ChargeBee prevents you from doing the biggest blunders in taxation, and also allows you to keep track of invoices paid through credit card or through bank transfer (and specify the Dunning rules accordingly). Overall, I'd greatly recommend it.