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.
Try downloading this driver for a generic USB-Ethernet cable, mine was available on plugable.com - NT-USBC.
Apple does not provide USB 3.0 gigabit ethernet adapters, but e.g. Edimax and Kanex work for Apple and have adapters made for MacBook Air and more.
You should be able to. RealTek - the makers of the 8152/8152B and 8153 Ethernet chips that are used by the majority of the generic USB connectors released an official, signed driver here: RealTek - Download Site for Generic USB 2.0 Ethernet Adapters for Mac/OSX
The drivers include support for 10.6 to 10.12 (Yosemite, Maveric, El Capitan etc).
These generic USB2.0 to Ethernet devices are sold under many brands including Agiler, Plugable Technologies, Belkin, Insignia, Sabrent, Amazon Basics etc. They often come in 2 forms a Short Cable Version and a Solid Body Version.