I am from Hungary and I would like to purchase sometimes from US web services like iTunes and so on. They require US cards and billing address. Is it possible to have one without traveling to the US? If not, how can I open a bank account in the US what information is needed?
Get a prepaid card.
Have you tried google ? Not asking this to be rude but since the first link for me gives instructions on how to get a US Card - the downside is that if you wish a debt/credit card they seem to require that you deposit $1000 :(
Another option that should work is to open an iTunes account with a US address and then add credits to that account with gift cards (you can find these at for instance ebay) - I have not tried this myself, but I know of a few friends that have done this for there apple tv to get US shows on them (here in Sweden).