My story has three offscreen rapes (one is onscreen, but it's not exactly shown), many deaths, mental illnesses (depression, suicidal thoughts, PTSD), underage sex, and incest. How to handle these topics appropriately?
I would recommend reading the work of other authors who have dealt with these topics in an appropriate manner to see how they approach them. For example, I have not read it, but Robin McKinley's "Deerskin" has had many positive reviews about the way she deals with rape and incest in the story.
You will need to research your topics, research what it's like to have those mental illnesses, and listen to people who have dealt/are dealing with those issues. Steer clear of stereotypes and clichés such as "princess finds her true love which fixes all her previous trauma". No. Just no.
Too many authors add rape/incest/many deaths into their stories as a substitute for character development or good plotting. Like, making the bad guy a rapist or making them kill lots and lots of people or making them drown puppies so that the reader will really know how bad they are. Make sure these topics are in your story for a good reason (or remove them if there's no reason for them), and you'll avoid that pitfall.
I agree with @Riorlyne. Don't romanticize these things, but also don't just throw them in there just so that your story seems dark and intense. Honestly, a talented writer can take just one of the things you listed and make it plenty dark to last the entire novel. There really isn't a need for so much trauma.
If your story is really that dark, make sure you have some regard for realism. Research, research, research. Look into how people react to certain traumas. Remember that trauma doesn't just go away because of love or overcoming or something. Trauma lingers for years and years –possibly until the day the character dies. It's not a fun subject to write about, nor to read about. Don't make things too graphic or too explicit.
Really consider why you've included all of those things in your story, and how many are actually needed for the plot/character development. I'm not saying you shouldn't include them, but just make sure they're actually necessary. These things aren't edgy or cool, they're extremely traumatic and horrible situations that should be handled as such. Just be careful and research, and I'm sure you'll do fine.
Two of the rapes I could take out, but one is important to the plot. Also, this is based in medieval times.
Take them out if you can, but if you can't, don't. Don't compromise the integrity of your story, whether it be through adding too much dark sh*t, or taking out too much of it. There's a really delicate line between grotesque and necessarily dark, and it's up to you to decide where to draw that line.
I agree with @writelikeyourerunningoutoftime.
If the setting requires a dark and gritty story, make it dark and gritty. I have a story that is substantially darker than my others because as opposed to being a journey/romance/adventure or a romance/friendship/war story, it's a mafia/corrupt city/betrayal/trauma recovery story. So the body count is about the same, since the cast is small, but it includes many MCs, if not all of them, as opposed to the others, which have supporting and major character deaths and many MCs get injured.