Does the Japanese Chin dog breed have eye problems? If so what are they? Do the eyes ever come out of the socket? I have a friend who had a ?

Like other brachycephalic breeds with protruding eyes and shallower than normal eye sockets, when some form of head trauma occurs, it is more likely than with a non-brachycephalic breed that eye proptosis (an eye that becomes displaced from the socket) will occur. This is NOT a normal everyday occurrence in ANY breed, including the Pekingese. Their eyes do not simply fall out without any cause. 

Yes as with any other brachyphelmic breed.

No, they are not PRONE to having any eye or eyes "pop" out of the socket.  Not all eye issues in dogs (or Chin) are inherited or congenital, some are acquired, via trauma, tumors, nerve diseases that affect the ocular nerve, or due to vestibular syndrome. 

"Chins" do have various eye issues (like cataracts, and strabismus [eye coordination disorder] ).  The "protruding eyeballs" can lead to corneal abrasions, (and subsequent eye ulcers) specifically due to the protrusion; all of which ......you could have GOOGLED.https://www.dog-learn.com/dog-breeds/japanese-chin...

why would people breed such a creature if that occurs? That's awful.

They look like cross eyed mutts. Maybe that's why they always look evil.