Setting up a US bank account for a non-US resident isn't all that hard. Worst case for most major US banks is that you might need to get a notarized copy of your passport and the application form, which can cost you a bit (first time I did it, years ago, the local US consulate charged me $65 for it).
Have you tried this recently? It's practically impossible, thanks to money-laundering paranoia. The closest I got was Western Union, but they said the account couldn't be used for any kind of internet commerce.
Yes, I have, with TD Bank. Didn't run into any problems at all.
If you run into problems, buy an off the shelf Delaware corporation online, with a registered agent, and create the account for that. Bonus, depending on where you do business you can potentially save quite a bit of tax by structuring things properly that way.
None that I've dealt with have required that. They do require to verify that you are who you claim you are, and without a US address you might find that requires extra paperwork (such as getting official proof of address and identity document certified by a US consulate or a notary public), and some banks might just not want to deal with the hassle. You also do need to fill out forms to establish your tax withholding.
I've held personal and business accounts with Chase and TD Bank, as well as trader accounts with a couple of brokerages, and the documentation requirements were pretty basic in all cases. I've looked into opening accounts at other banks at various points too.
In some cases the easiest alternative is to simply buy an off the shelf US corporation, and have the company that sets it up register a US bank account for it. That's certainly often less paperwork and hassle than dealing with a personal account, especially if you set up a Delaware corporation.
That's a good alternative, but then I'd have to pay accountants and lawyers and the rest... It does give you flexibility in other areas, though (such as, I will finally be able to open a Stripe account!).
You likely have to have identity documents certified, that's about it. A visit to a US consulate (check first if the provide the required services) or notary public should be able to help get the right stamps and/or seals to satisfy them.
