From my experience, cats are finicky and will eat mostly meat and fish given the chance. They eat grass to help them puke, which they do far more often than DF models.
Dogs will eat anything that smells like it might not kill them. My dog eats snow peas, lint, her blanket, hair, dog food, cheerios, toilet paper tubes, grass, other dogs' poop, etc. I agree with dogs eating fat, chunks, tallow, and so forth. I think dogs should even try out eating stuff like charcoal once or twice before realizing it isn't good to eat. All puppies and unattached untrained dogs should also chew on unattended leather/cloth items not kept in containers.
Typically pigs (strangely absent from DF?) are fed scraps deemed inedible by people. So it might be appropriate for a finished meal to give out a seed/bone/shell if appropriate and also a little purple "food scraps" that opportunistic scavengers will eat. Including vermin. Or dwarves will toss them into a nearby Trough and animals will hang around areas with Troughs in them.
So maybe something a little like-a this:
Scavenger: Looks for remains, food scraps, and hangs around refuse stockpiles and Troughs.
Herbivore: Chomps down grass tiles (clear down to the clay), shrubs, and in some cases denudes a tree (so it looks like a leafless winter tree). Hangs around grassy parts and browses nearby Troughs.
Carnivore: Kills vermin (even bears will snack on a squirrel or something given the opportunity) and small animals. Will eat unrotten animal parts and rotten ones if starving.
Also I'd like to point out that most livestock these days is fed stuff that's not fit for human consumption. And it's been like that for a while. So food parts that don't show up to dwarves right now like the chaff and stalks from cave wheat would be fed to the animals. You could have "fodder" appear when food is processed in a farmer's workshop.