Does anyone know if this is indeed possible and what kind of things I'd need to show and provide in order to get it done?
I do have lots of family and friends all over the US, mainly the larger cities etc so using an address of one of them shouldn't be too much of a problem.
Could I pop into a local or even one of the large US banks with my passport, birth certificate, drivers license etc as proof of ID and open a basic account with them?
I don't know how hard it is in the UK, so I can't say having an account based here is easier or harder than in the States. I do know, having opened and closed many accounts over the years due to my semi-nomadic ways, you can open an account in the states very easily. I've only heard of two people in my whole life who have ever been denied an account at a bank (one because he had written a ton of bad cheques, the other because of some problems with probate after a relative died).
Anytime I've opened an account, savings or chequeing, I just needed my ID and some money. I may have had to show proof of address, but that is relatively easy to obtain in your intended US address. I don't even know if you always need that because I know I had two accounts at college, both opened well before I had a chance to establish an address. However, they may make exceptions for college students and they were both savings not chequeing.
If you do decide to get the US account, I'd just have the people at the address get some sort of mail put in your name, at least temporarily.