Company in US, Driving License
Yes, you can register a U.S. company as a Hungarian citizen.
You will not be required to have a U.S. address or social security number, nor will you need to submit any financial or identification documents. We have a research article titled 'U.S. Companies for Foreigners', which is very helpful in providing resources and direct application links, as well as comparing the different entity types and top recommended states to register your business in.
In regards to obtaining a U.S. driver's license, that is something you would need to consult with the Department of Motor Vehicles, in this case, Nevada's DMV offices specifically.