Are there any kabbalistic or hasidic explanations given to the question of theodicy? Beyond “rational”/philosophical explanations, are there any mystical reasons given to why bad things happen? If so, what are they, and where could I find them?
One of the big chiddushim* of Kabbalah in theodicy is that both good and evil serve Hashem. When discussing the Yeitzer Hara, Rabbi Shimon tells the Chaveirim something that "they had never heard before": that it was necessary1.
Kabbalah teaches us that nothing exists that is not some form of Hashem's self-expression, so therefore evil must be a non-existence2, and simply the absense of light.
A mashal is given that a man is walking in a building and hears a woman screaming in agony. He is convinced she is being assaulted so runs to her rescue. When he bursts into the room, he realises that he made a big mistake. The room is actually a birth center, and the woman is screaming because she is in labour.
From the above example, we can see that the supposed evil was proven to be goodness, and it happened in a split second of clarity, or light.
However, Kabbalah also teaches us about a type of evil that is indeed a creation3, and this is the result of the שבירת הכלים. When this happened, many positive traits and attributes fell into the wrong places. Now it is possible for people to get actual enjoyment in evil activities.
Chassidus teaches us that our job is to rescue these "sparks", in two ways: through avoiding temptation and doing teshuva, as well as observing the world through the lens of Torah in order to learn from it to improve our character and reach higher levels. I'll give some examples4.
In terms of the former, when we avoid a temptation, we weaken that temptation for everyone on the planet. We keep on doing that until there is no temptation left. If we succumb to the temptation, we are able to even more powerfully rescue the spark, by doing teshuva, because we experienced the energy that is locked away in the sin, and are able to more directly apply it to goodness. Here's an example:
A husband and wife love eachother to the maximum capacity of love. One day they have a bitter argument, and there is a lot of anger. Eventually they "do teshuva" and make up. They find they actually love eachother even more than before, and more passionately than before. This extra love capacity, and quality of love came from the passion of the anger (the passion of which is neutral), which has now been "rescued" from said anger, and now belongs to the love.
For the latter, an example given sometimes is Hitler y'sh. Hitler had a lot of energy and insanity for evil. The only bad word in that sentence is "evil", and if the "energy" and "insanity" could be rescued for good, that would be wonderful. If we could have someone arise and lead the world towards "insane" goodness and Hashem, with the same energy as his opposite, then we truly will have rescued some powerful sparks for goodness, before evil is banished forever.
Chassidus teaches an idea of "descent for the sake of ascent"5. It teaches that Hashem's goal in creation is to have a home in even the lowest world; the world of sin and death. By the sin of Adam Harishon, we integrated this שבירת הכלים into our souls, and are thus now able to partner with Hashem in our own "fixing" of the world. Hashem wants us to earn our good and gain true reward6, which brings us back to our Zohar: the Yeitzer Hara is necessary, and our working on that, performing mitzvot and learning Torah is necessary in order that we should fix life itself, for Him, so that we can build Him an eternal home in this lowest world - His deepest desire.
As zunior rightly said, we don't know why great evil takes place in specifics (although we do know it is from Hashem and only Hashem - what other source is there for anything?), and we don't want to know. We don't want to hear anything that justifies it, because we accept it is not justified. The only thing that Hashem asks is that we protest it, we fight it, and we put a stop to it once and for all ourselves, and maintain that Hashem is good and there is ultimately only good. One day we will be raised to the level of awareness of Hashem and our tears will be wiped away, but don't we dare get complacent in eradicating evil from this world with every fibre of our being, and bringing in as much Godly light of goodness that we can.
* this term is used loosely. There are of course older sources that root these ideas, but they were developed and made more prominent by the rise of the Zohar etc.
1 - Zohar Bereshit 16
2 - See e.g. Emunot V'deiot 1:3, Tanya 12 etc.
3 - Based on Yishayahu 45:5-7
4 - See this shiur for an in-depth treatment of this subject by a prominent educator of chassidus.
5 - See מאמר באתי לגני תשל"א for a full discussion of this topic.
6 - See Derech Hashem 1:2, and the above (footnote 1) quoted Zohar passage
According to Kabbalah the root of all evil is the שבירת הכלים. This the source of the "Husks/Shells" which are called cruel oppressors.
You will see this referenced at the beginning of most Lurianic books (take for example שער הכללים) However you wont find an explicit answer to your question there because these ideas were meant to be contemplated and the real truth is a סוד.
Beyond that you will certainly find many sources that talk about things like Sexual Immorality causing suffering but the ultimate source is what I mentioned above.