Most male dogs are happy to be able to lick their own dick. Imagine how happy the dog would be with two dicks and be able to lick them both! Man, he's a happy guy.
That sonuvabitch must have just humpped the neighbor's cat. Look at the silly fucker, he's happier than a dog with two dicks.
Means that something is very fine, or that you feel excellent. If someone asks how you feel and you simply say "fine", you might just be saying it out of convention. Adding the "dog's hair split three ways" shows that you actually mean it.
Another form of the expression is "finer than a frog's hair split three ways", although literal frogs do not have hair.
You: How are you doing today?
Me: Why, I'm doing finer than a dog's hairsplit three ways!