Wow this has gone south quickly.
The law on cats works both ways - the owners can't get damages off you, and you can't claim for car damages.
Dogs it's different and depends on the circumstances - dogs should be on lead on public roads, so if it's offlead and you hit it it's not your responsibility, however you do have to report it - but if it's on the pavement and you clip it, just like humans, you could be liable for vets bills.
Interestingly, you actually by law should not serve or brake to avoid hitting wild animals - or cats! if you swerved and caused an accident you are at fault for hitting them, you should just continue by rights.
Ultimately, if you let your animal behave like a wild animal, it has the same chances of being injured as a wild animal, simple as.