Can older iPhones receive Animojis created and sent from iPhone X? iPhone 7+ (iOS 11.1) owner here. If a friend buys an iPhone X and creates/sends me an Animoji, how will it appear on my phone (if at all)?
Short answer:  Yes.

According to Apple:


  You can create your own Animoji and share it with anyone using an iOS
  device, Mac, or smartphone.


The iPhone X can create the Animoji which can then share the animation to anyone running iOS, macOS, or a smartphone - ANY smartphone.  See this article for more details

The Animoji is saved as a .mov file which is sent via MMS and can be viewed by virtually any smartphone (not just iPhones).