No. you cannot get a carrier plan without a Driver's License and Social security number, and you cannot get either of those if you are not a resident of the U.S.
If you are going to use an iPhone in romania, why would you buy one that is locked to U.S. carrier like AT&T? Phones locked to a carrier cannot be used with another carrier.
I think you should perhaps think about what it is you are about to do, and what it entails.
Even if you could pay the AT&T plan abroad, you would be incurring in Roaming charges every month with AT&T since you'd be outside of their service area.