Do catholics consider the problem of theodicy (Why God allows evil on this world) an open question, or is it "solved", seeing that so many people have tried to answer it. If it is not solved, what is the reason?
Yes, it is an open question.
Yes, it is a "solved" question.
No, it is not an OR question.
The question you are asking "Is Theodicy an "open" question or a solved one?" is based on a faulty premise. Namely, "open" does not mean "unanswered" or "unsolved" - it means one that is continually being investigated. Thus, the opposite of an "open" problem isn't a "solved" one, but rather an unasked one.
At best, one can ask, "Is there a consensus position?" and the answer is "within a fairly broad spectrum, yes." Yes, there are several "solutions" that theologians have posited that each have a significant consensus behind them. Go find them - that's a book.
In order to highlight the faulty premise, let's use another question as an example: "Did God create the world ex nihilo or did it evolve?" As contentious as the question is, no one can say there are no "solutions" to it. Non-theists have posed several answers, Theists have posed several answers, and there are even significant groups that seek a "theistic evolution" answer in the middle. Is it an open question? Sure! People wrestle with the origins question everyday. And is it a solved one? For many people, again, the answer is "Yes!" They have weighed the evidence and personally picked the "solution" they feel is most compelling.
In theology, there are very, very few "closed" questions. John Dominic Crosson believes the literalness of the Resurrection to be an "open question," despite Paul's very obvious answer to "If there is no resurrection, then we are to be pitied above all men!" The closest thing I can think of as a "closed" question is the literal existence of a King David. For many years, theologians thought it an "open question" as to whether or not David existed. When archeologists found a steele mentioning the House of David, the evidence pretty thoroughly closed any further debate, because it is really hard to argue with tangible evidence to the contrary - and yet there are still some who claim it is "open"
Back to the "problem" of theodicy - or "Why does a Good God allow evil?" There are several solutions to the problem - but that does not mean it is unasked. It is thus open and solved at the same time. And frankly, until there is no more evil, or until people see God as not good, it will always be asked. It will and must be open to new queries as long as man asks questions.
I'd like to post more or less a paraphrased pseudo-lazymans synthesis of Saint John Paul's Letter Salvici Doloris, which got me through a secular university's class on Job swimmingly.
First of all, the answer to the problem of evil is tied with the meaning of suffering. If it weren't, it wouldn't matter whether it was evil or not, I think so Salvific Suffering (as the pope put it) is pretty much the answer. But there is a lot more in the letter.
Pope John Paul II pointed to St. Paul's message that his suffering makes up what is lacking in Christ's to show that all suffering points towards salvation. And Job, prefiguring Jesus, as well as the Suffering servant in Isaiah, who do nothing wrong and yet are tested and broken show that suffering is not merely retribution. Jesus even says its sometimes rather arbitrary when talking about the tower that fell on some guys. But, even of it arbitrary, its not outside God's will. It's just God's passive will in action. The things he permits to happen by removing his hand (see Job 1 or 2) rather than my sticking his hand in the pot (see the 2nd to last Chapter of Job).
St. John Paul says the answer belongs to the class of problems tied up with the mystery of the incarnation (for God so loved the world...) and that the answer to it lies in Gods love rather than his wrath. Because, even though stuff is tough, we are always under God's close and personal gaze, we are not in Hell, and because of it we should expect that any grace given, any supernatural truth that manifests itself in our lives, should be seen as a direct sign from God.
It can never be fully solved because God is in the equation, and that is why it is also referred to as the mystery of evil.
This article, Evil | New Advent, states the problem as,
To what, then, is the evil of human life, physical and moral, to be attributed as its cause? But when the universe is considered as the work of an all-benevolent and all-powerful Creator, a fresh element is added to the problem. If God is all-benevolent, why did He cause or permit suffering? If He is all-Powerful, He can be under no necessity of creating or permitting it; and on the other hand, if He is under any such necessity, He cannot be all-powerful. Again, if God is absolutely good, and also omnipotent, how can He permit the existence of moral evil? We have to enquire, that is to say, how evil has come to exist, and what is its special relation to the Creator of the universe.
Herein is the difficulty, evil in the presence of an all-good, all-benevolent, all-knowing and all-powerful God, and from scripture, who himself hates evil.
The article continues
The solution of the problem has been attempted by three different methods.
Therefore answering the question, it is clear that the problem hasn't been fully solved though three solution have been proposed to attempt to solve it, and perhaps, the problem will never be fully solved at least this side of heaven.