European business registering in USA
If I open up the business to buy from and ship to end users not only in EU but also USA and other continents, will I be able to register the business in USA, even if I keep stock and ship from Europe?
Our specialty is helping people with questions regarding formation of US entities, so I would not be the right person to give any meaningful information on forming entities in other countries. However, from your description my logic tells me you would need to register for business in Europe this way or another, if most of your business activity will take place there.
You can always register a US entity to serve as a branch for your US activity, but I don't believe you can do business in Europe without registering it there.