Does ElGamal authentication exist? Are there any natural ways to transform ElGamal encryption system or ElGamal signature scheme into an authentication protocol?
You can turn any public-key encryption or signature system into an authentication protocol, and that includes both ElGamal encryption and the ElGamal signature scheme.

Protocols for this can be found in most textbooks. E.g. 10.3.3 in Handbook of Applied Cryptography.