All I can remember is a bull terrier who laid on his side and when he breathed in, his legs would go up with each breath. There was two children. The boy had longer hair that flopped when he ran. The animation kind of reminds me of The earlier simpsons

This is driving me crazy... please help

Hmmm? It might be Family Dog, a short run primetime cartoon series on CBS.

YouTube - Family Dog - Intro

http://en.wikipedia.org/wiki/Family_Dog_(TV_series)

Here are some links.