Can a service dog fly in the cabin of a plane for international travel?

That depends on the country of departure's laws and the country of arrival's laws. Some countries require all dogs, including guide and service dogs, to enter the country as cargo. Check Consular Information Sheets for the individual countries involved to find where to check the regulations for dog import/export. See the link below for Consular Information Sheets.
Yes, on flights originating and arriving at points in the U.S. or U.S. territories. The Air Carrier Access Act requires air carriers to permit service animals to travel in the cabin with their disabled handlers unless an animal physically does not fit in the space allotted to the traveler and there is no other space on board to accommodate them. If the animal will not fit in the cabin, the air carrier must offer to transport the animal as cargo free of charge.