Can I fly in the USA with an FAA ATP, if I don't have a commercial certificate?
Maybe. Your ATP certificate does give you commercial privileges, as per 14 CFR 61.167:


  (a) Privileges. (1) A person who holds an airline transport pilot
  certificate is entitled to the same privileges as a person who holds a
  commercial pilot certificate with an instrument rating.


But, that doesn't authorize you to work in the USA or even to enter the USA, which is an entirely different question about immigration law. As a foreigner, you'll need an appropriate visa or a green card in order to be legally employed and paid. You can ask more about that on another site.