no, but the family dog goes missing, and is seen eating chocolate, but otherwise is okay as of s4. bill stabs a street dog with a syringe, but its unknown if its dead.
one of the murphy's neighbors, goomer, is known to always watch the family and is obsessed with frank, wanting to be him. there's an episode in which he sneaks into the home and dresses as frank.
yes, frogs and birds. you may want to skip s1e2 18:00-18:05 seeing as one of the kid accidentally eats a bird. it's brief and isn't gory, but you can see blood around the kid's mouth.
i believe frank definitely has some form of mental illness, from being abused as a child and being drafted into war. he is seen frequently yelling at his kids, as well as others. i haven't watched the show in a while, so i can't remember any physical violence, though.