Is commercial domestic service using foreign registry aircraft allowed in the USA?
Yes, it's allowed under both part 135 (on-demand) and part 121 (airlines); 121.153 and 135.25 both have the same wording:


  A certificate holder may operate in common carriage, and for the
  carriage of mail, a civil aircraft which is leased or chartered to it
  without crew and is registered in a country which is a party to the
  Convention on International Civil Aviation if— [...]


And US pilots may fly foreign aircraft within the US.