At least in EU, nope! Because it's not just "automation". It's more about dealing with other people's money, accounting, taxation, sensitive data collection etc.,etc. You should be first registered as a "payment processor" type.
You can charge customers by direct debit (bank to bank) with an API. You can create your own pdf invoices and email them. You can charge customers with paypal or use credit cards or any of the EU specific online payment methods.
There is no EU directive or law prohibiting this. You don't need any kind of "special data collection" authorization. Most SMBs just print their own invoices with MS Word or some generic software package.
That's incorrect, at least for some countries in Europe, including Italy (which you mention below).
You can have your in-house billing system (but all the other eu crappy regulations apply, eg. GDPR, VATMOSS), what I think you are referring to is becoming a money transmitter, which requires a special license.
