Dogs do die, the wild dogs from the hunters that adult Bambi fight off are killed when Bambi makes rocks fall onto them, they aren’t seen moving so they were likely crushed, there are also some wild dogs that Bambi headbutts and sends them flying and they aren’t seen moving again
Near the end of the film a pheasant is shot by a hunter - her body can be seen falling out of the sky and hitting the ground. Although Bambi's mother is also shot and killed earlier in the movie, her body is never shown.