Yes, they do. In fact, the Green, Gold and Platinum have exactly the same underwriting criteria (based on circumstantial evidence on internet forums), so you can in theory get a Platinum with zero credit history.

Not at all. In fact, based on my own experience, it is the easiest bank to deal with. I opened a bank account and credit card ($5k credit line) with HSBC when I moved to the US in February 2013. In May 2013 I got the Amex Delta Gold with an $8k credit line. I had no credit score at that point, as one needs at least 6 months of credit history to generate a credit score. I did have to send in a payslip, but that was about it. As long as Amex believes you have the means to pay it off, you can get one of their cards with basically no credit history.