You are absolutely, 100% correct and the sad thing is he may have already done some damage. It's around this age that any children figure out that life is not all roses, and that people can be mean, get sick and even die. It's so very important to be careful what they are exposed to on a daily basis and important to think about the things we would notice, as well as the things we might not necessarily notice and I will explain. The obvious things are movies and television. All movies have a rating, and they have them for a very good reason.
The American Academy of Pediatrics says
"The movie ratings of the Motion Picture Association of America (MPAA) are often poor predictors of the amount of violence. Many PG or PG-13 movies are as violent as movies rated R by the MPAA. A review of 98 PG or PG-13 rated movies found several were full of violence, containing as much or more violence than many R rated movies.
One example is the 1994 PG-rated "The Jungle Book." It contains nearly 100 violent acts. In a study from the UCLA School of Public Health, "The Jungle Book" was found to be one of the three most violent movies available. The other two, "Time Cop" and "True Lies," were R rated. The average number of violent acts in R, PG-13, and PG movies were similar: 62 for R rated movies, 55 for PG-13 movies, and 56 for PG movies."
The only movies she should watch at this age are rated G. These ratings should be followed implicitly. If she walks in the room while you are watching something (movie, the news, television shows) then you should turn it down, off or change the channel until she leaves the room. This is hard for most people, they have the tv on all the time and never really stop to consider what's on it. Your ex is watching TV and spending time with his daughter, a pretty normal pastime for families, and who wants to watch a rated G movie? It's all about self control as a parent and remembering that it's not about you, but about the child.
I don't even like for kids to watch cartoons because some of the commercials are directed toward children, and its just not good to bombard them with all of that stuff. I always suggest that parents start a dvd collection with age specific dvd's. If you want to watch an adult movie, sometimes you have to wait until she is asleep. Even when she wants to watch something, you have to remember that she has no idea whats good for her or bad for her, so you have to make the decision, not her.
The news is another no-no, children cannot differentiate between what's real and what's not real. When they see movies, its real to them, when they see the news, it's real to them. For example, when 9/11 happened, the news replayed the falling of the towers over and over again. Many children thought that many, many buildings fell, not just two. They also though this was happening every day for weeks and months since it was shown on tv for weeks and months after the event.
Aside from being careful of what she watches on TV, be careful what you talk about around her. This is the age where she can take in more info that she probably ever will in her life, she is literally a sponge. The problem is, she hasn't gotten a complete grasp on the subtleties of the English language. When you say "so and so is dead meat", or 'I'm going to murder that clerk", she may take simple expressions very literally.
Video games, same thing, they put these ratings on the games and movies for a reason, and the safety of our children depends not just on the ratings, because as I mentioned, the ratings may not always be correct. The best thing overall is to try the game yourself, watch the movie yourself, listen to the music yourself, before you expose her to it.
Now most parents will dismiss all of this because its inconvenient to do whats best when you can just plop her in front of the tv and tell yourself it doesn't matter. It's much harder to be stringent about what she watches, change the channel when she walks in the room, or play Disney cd's in the car. Raising kids requires a ton of patience and a ton of self sacrifice. Not everyone understands it or are willing to do it. I would suggest you keep talking to him and keep trying to make him understand. You might get through at some point. Additionally, I believe if you have court ordered custody that you may want to talk to the mediator or if he forces you to, the judge about making him stop exposing her. I can't say that they will but it might be worth looking into.
I'm including some links below on the possible negative effects of ignoring child ratings.
Please let me know if I can be of further assistance