Is an alternate airport always required when flying internationally?
As a simple anwer, no, it is not always required.
ICAO Annex 6 4.3.4.3 Destination alternate aerodromes

For a flight to be conducted in accordance with the instrument flight
rules, at least one destination alternate aerodrome shall be selected
and specified in the operational and ATS flight plans, unless:
a) the
duration of the flight and the meteorological conditions prevailing
are such that there is reasonable certainty that, at the estimated
time of arrival at the aerodrome of intended landing, and for a
reasonable period before and after such time, the approach and landing
may be made under visual meteorological conditions; or
b) the
aerodrome of intended landing is isolated and there is no suitable
destination alternate aerodrome.

Also, there are similar descriptions for the requirements for take-off and en-route alternates.
EASA adds aditional requirements in OPS 1.295 in the form of maximum flight time of 6 hours and at least two runways at the airport, but it is still allowed. I can only assume FAA does a similar thing