All the animals would be able to talk to eachother. Just at the start have only house pets, and then as the rp progresses add more and various animals. As for prey-type animals like Rabbits and snakes and mice, at first because the pet type animals are so accustomed to dog/cat food and pellets they wouldn't think much about hunting. As the world becomes more and more 'uncivilized' then it would be more so that the prey animals would turn into prey as the house pets would move farther out to the wilds. 

As for starting I figured that they would live in the same neighborhood and it's not until their owners are evacuated and all the pets haven't eaten in a while start to wonder what's going on exactly. If it's a natural disaster it would be more apparent of the danger as the humans get rescued and the pets are left behind to fend for themselves. 

It depends on the kind of doomsday everyone would be up for.