What are the requirements for an airport to be designated as an international airport in the US? Is there a minimum runway size or required customs office?
There is no US regulation about whether an airport can be called "international." The Secretary of the Treasury designates the official list of international airports of entry. But not all airports on this list are even called international, and it does not include all airports with international flights. There are even some airports called "international" with no customs facilities or plans to have them in the near future. The airport name is determined by the airport owner.
It is defined by the ICAO in this glossary:
International airport. Any airport designated by an ICAO Contracting State in whose territory it is situated as an airport of entry and departure for international air traffic, where the formalities such as customs, immigration, public health, agricultural quarantine and similar procedures are carried out.
"International" is supposed to be a code word to inform pilots and ATC that the Border Guards have a presence, and can admit you into the country: stamping passports, collecting duties, all that stuff.
What you shouldn't do, say, is fly from Canada into Oswego County Airport, rent a car and drive over to Syracuse Hancock International to clear Customs/Immigration.
You are to land at Hancock International, clear Immigration, then if needed fly onward to Oswego County.
The only exception I can think is if you have an emergency, such as weather unexpectedly closes in and no international airports are viable. Aviation authorities will be sympathetic; they want you safe. The border guards will be less amused. They will suspect a setup and that you are up to no good. From their POV, if conditions were marginal, you shouldn't have attempted the trip.
You wouldn't want to make a habit of it.