I'm going through the bank opening process as we speak with SVB however I'm from the UK and with a new branch of SVB being opened in London, I feel the OP will be in a different position as he isn't from the UK.

If you don't qualify for an SSN you can get an ITIN (Individual Taxpayer Identification Number) that can be used for some things in place of an SSN, but it's not a "first class" identifier. I think an ITIN is fine for setting up a bank account, but it's been a while since I did it.

Yes, you don't need an ITIN either. A passport and another (any country) credit card with your name on it will be enough.

People on a US student visa often aren't required to file US taxes and therefore do not need an ITIN.