Is it legal for a pilot with an FAA license to fly a foreign registered aircraft within the United States? Is it covered by FAA regulations, or the country of aircraft registration?
Yes. 14 CFR 61.3(b) covers this (my emphasis):
(b) Required pilot certificate for operating a foreign-registered aircraft within the United States. No person may serve as a required pilot flight crewmember of a civil aircraft of foreign registry within the United States, unless—
(1) That person's pilot certificate or document issued under §61.29(e) is in that person's physical possession or readily accessible in the aircraft when exercising the privileges of that pilot certificate; and
(2) Has been issued in accordance with this part, or has been issued or validated by the country in which the aircraft is registered.
So a US license (i.e. one issued under part 61) is allowed as an alternative to a foreign license for operating the foreign aircraft. The other country could have rules on this too, but I have no idea if they would be enforceable (obviously, one country's laws are usually not valid in another one).
There are reciprocity treaties requiring most countries to allow foreign aircraft to fly in their airspace. It probably depends on whether the country involved has such a treaty in place with the United States.
In general, the FAA tolerates foreign aircraft to be operated in the US indefinitely so long as they meet airworthiness standards. In theory, they claim you have "imported" the aircraft and require you to re-register it, but this does not normally happen.