Hey guys, sorry I've taken so long to reply! I started my new job on Wednesday, excuses, excuses... And as luck would have it, my green card arrived on Wednesday also! I'm hoping that the letter it came with will be good enough proof of where I live as it's an official government letter. We're going to try again at the bank tomorrow and *fingers crossed* they'll let me open an account.
Even though the immigration officer told me that my Alien Registration number is "basically" my SSN, I did some research and found that not to be the case and that I could apply for my own SSN. So I went to the Social Security Office on my day off and they're sending me one out in the mail.
It's been a stressful 2 months but hopefully this whole palaver will be over with soon. Thanks to everyone for your help and advice, I did read them all at the time, I'm just terrible at getting round to replying! Sorry!
Now to get college courses and finance sorted!