I never heard the dog story on cable, so maybe it's edited out. I did "watch the movie" too.

I'v seen some broadcast versions that have cut the dog story out.
I will not be pushed, filed, stamped, indexed, briefed, debriefed or numbered. My life is my own.

Try watching the movie.

I think the dog story served a couple of purposes.
First it was essentially Paden's moment of truth. I think he believed that letting the dog die would put him on the path to becoming like Tyree. It was the moment where he had to stop being an outlaw.
It also helps display the differences between Paden, Cobb, and Tyree. It wasn't entirely necessary since the movie makes it pretty clear. Tyree is just mean without any real reason. He'd kill for no reason and doesn't really like anyone. Cobb kills for profit. He feels neither pleasure nor remorse about killing (note he finished his coffee after killing Paden's predecessor). He genuinely likes Paden, however. There's anger in his voice about Paden choosing to help the dog. He couldn't care about the dog, but probably wouldn't have cared enough to shoot it either. Paden was still capable of caring and had to help the dog in order to help himself.

I like the dog story, because it adds to Paden's quirkiness and unpredictability.
But it wasn't really necessary, for pretty much anyone to realise that he cared about Stella.

Did you miss the story that Cobb tells Stella about Paden and the dog? He tells her that there is no telling what Paden will care about and that a dog had started following there gang. One day, they're trying to escape a posse and the dog trips up Tyree's horse and Tyree shoots it. Paden jumps off his horse and picks up the dog (which is still alive) and tells them to leave him. He ended up getting arrested because of the dog. (Which Cobb then goes on to say that Paden didn't even like.) Hence, all of the dog references.