I make most of my MP shopping purchases with an Amex, so yes, you can use other cards without a problem. Cartera has gotten significantly better at responding to missing miles inquiries IME, though you should wait ~4 weeks from the time of purchase to file a request if they still haven't posted.

IME, it's the retailer who ends up dictating how long it takes and whether or not the miles post first time around without an issue, vs. the card.