Hah, the first album - yes, vinyl - that I begged my parents to buy me was the soundtrack from the play "Hair" - quiet the Vietnam, hippie era play complete with stage nudity and all- with overt references to just about everything. I think I was about 10 and I know I didn't get most of the references - I just liked the music. Oh, and they did buy it for me - we were all so very liberal then. LOL
Anyhoo, I find that all Disney movies, and many traditional children stories, have darker elements. Bambi's father, Nemo's mother, the ostracization of the Hunchback, the evil stepmothers, Simba' father, etc. Don't forget that the Grimm's Brother Fairy Tales were originally written for adults - it was not until later that they became childrens stories.
I usually avoided these types of movies when my son was very little. When we watched TV we watched PBS mostly. The others scared him - and really, what was the point of having my 3 year old son have nightmares after watching a "childrens" video.
I think you and your hubby are doing the right thing by screening movies before your daughter sees them. It will not hurt her to miss Bambi - I thought it was a terribly scary movie when I was a child.