Surprisingly yes a dog does die in this movie. Bambi pushes 2 dogs off a cliff towards the end of the film. It can be inferred that while the other 2 hunting dogs ran away these two stopped making noises and were not shown after being knocked down.
Not abandoned directly, but the Great Prince often leaves Bambi alone for the day while he goes patrolling, then later gives him to a doe to raise him. The Great Prince eventually realises how much he loves Bambi and doesn't leave him alone anymore.
Bambi becomes unconscious after falling from a cliff after escaping the dogs. The Great Prince and Bambi's friends think he's died, but Bambi wakes up shortly after.