Once upon a time, fables and fairy tales were not analyzed for how much trauma they would cause, and the phrase age-appropriate did not exist. In fact, Disney’s tales are, in generally, terrifically sanitized versions of other stories.
In the original Cinderella, the stepsisters cut off their own heels and toes in an effort to stuff bleeding feet into the shoe, and in some versions, the stepmother and stepsisters were killed, or at least exiled at the end. Of course, this comes from a time when these stories were told to everyone, not just to children, and many of marriage/sexual awakening themes in particular were cautionary tales for adolescents.
However, children grew up with all sorts of violence in their stories, right up until these past couple of decades, and by and large, did not grow up with uncontrollable fears of stepsisters, talking flounder, or orphaned deer. They did, however, grow up with substantial metaphorical moral instruction, for better or worse.
I think, more important than -whether- there is violence are two other questions: how -graphic- is the violence (in this case, not very; iirc, the hunter and mother are only seen in silhouette), and what -message- does the violence send. Going back to the
Cinderella story, it teaches us two basic things: Being desperately greedy for something (someone) doesn’t mean we’ll get it, no matter how much we hurt ourselves or others in that pursuit; And, that the meek and subdued today may be in a different position tomorrow, so perhaps one should think twice before kicking someone just because you can. On top of the morality messages, it has an optomistic miracles-can-happen, don’t give up hope even in the darkest hour message. Not too bad as messages go, in my book.
It’s been too long since I’ve seen Bambi for me to really remember what themes or morals you might get from it, and it doesn’t really matter what -I- think anyway, in this case. It matters what you, as the parent-in-question think. I encourage you, however, to dredge up your schoolday skills of articulating the moral of the story, and deciding on that basis more than on trying to remove all the possibly ‘disturbing’ stories. Issues of violence and loss are real, and are best coped with by understanding them through fiction first.
OTOH, a lot of pop-culture violence basically teaches us that those that hurt us do so because they are Evil ™ , and we can solve this best by killing them, or in this more sanitary decade, by confronting them with weapons which will then get dropped for a fistfight, at the end of which the bad guy will be taken away by the authorities for properly administered punishment. I don’t think these shows get any better, from the point of view of an influence on children, by making them bloodless. Violence-for-violence-sake shows are not really helping us understand anything, and do little other than inure us to violence.
Err. Did I go into lecture mode again? Sorry, but you did ask! 
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