Italian-based US person doing business in New York
Can I create an American company in New York? I have a DBA, what I am told is that I need a LLC.
Can I make American invoices in Italy? Of course I will pay all taxes in New York.
You can certainly register any type of business entity in any of the US states. As far as invoices go I wouldn't be able to answer that question as I am not familiar with Italian laws.
To better answer your question I would need to know what kind of business this would be, who will be your partners if any, and what your target market is.