Dr. Dolittle starts with a flashback to Doolittle's childhood when he carried on frequent conversations with his pet dog, voiced by Ellen DeGeneres. The dog bites someone, thinking she's protecting her master and Dolittle's parents send her to the animal shelter. Animals that bite don't get adopted back out, so...
In the Eddie Murphy version he was sniffing people. Specifically, his future principal's behind, as dogs do, on his dog's advice. Hence his parents decided he should be separated from her.
Rodney the Guinea Pig is not seen or mentioned in the sequel. It's possible he died of old age.