Do I have to use the Apple USB Ethernet Adapter for a MacBook Air? Or can I use a third party one that is half the price? If there will be compatibility issues then I would prefer the Apple, but if it's just the same (like not needing to buy a SuperDrive any USB DVD reader/writer will do) can a third party one be used?
You don't have to but your mileage will vary between the adaptors if they work or not. The Apple ones have drivers built into OS X, most of the others should work but there'll be varying levels of success.