Are all animals called by summoning jutsu able to speak?
Although they didn't explicitly show whether they can talk or not, but one thing is for sure: that Pakkun can talk.

Kiba's Akamaru can't talk, so we can assume that other dogs of Kakashi also can't talk (since there is no evidence in the manga or anime).

So in order to answer your question: "Is there a rule that every animal can talk, or something like this?" I would say "No, it is not mandatory for animals to talk."