1. This seems like it would be more of an action, but yes. If you could make different textures, patterns, sizes and shapes, it might work.
2. They do have a twig item, although it would be cool to have it in the corner of the mouth.
3. So, as in carrying the bones and stuff back to the places like camps and dens? Yes, this could work, but you would have to tweak all of the sizes around. You wouldn't want a small warrior cat with a bone two times the size of its mouth running around.
But yes, your ideas are pretty good.