What does Wiesel think of theodicies? What is their answer to the problem of evil? If God created a world without evil, what would it look like and how would it be different than the world we live in now?
Please keep in mind that you have asked a true "opinion question," meaning that because Elie Wiesel never spoke specifically about theodices in Night, any reader must guess what the author "thinks." This being said, I am happy to share my thoughts.
Let us begin with one of Elie Wiesel's coveted prayers that can serve as a good introduction to a discussion on theodices:
Master of the Universe, know that the children of Israel are suffering too much; they deserve redemption, they need it. But if, for reasons unknown to me, You are not willing, not yet, then redeem all the other nations, but do it soon!
This shows (very well, I might add) the believe of the Hasidic Jews that a relationship with God should be personal, but is always welcome to include doubt. Just because there is anger or doubt towards God doesn't mean Wiesel is denying his faith. (In this way, I often disagree with many scholars who discuss Night as a book about a Hasidic Jew who loses his faith in God.) Put simply, experiencing the Holocaust first hand could not possibly be reconciled by a simple prayer. Now let's begin with the specific theodices that you mention.
1. Limited Theodicy / Satan as the Cause of Evil
In your explanation above, the key to this theodicy is that "this is a limited value interpretation of evil, because it does not answer why God allows evil and the suffering it brings, as Satan is under the power of God. Thus we wonder why God allows Satan to continue existing." And, ironically, although Wiesel doesn't mention Satan at all, this is precisely what Wiesel struggles with under Hasidic Judiaism. The struggle exists BECAUSE this theodicy is a limited interpretation! Humans, then, are left to doubt and argue. This describes Wiesel perfectly. For example, look at the following quotation:
Behind me, I heard the same man asking:
"For God's sake, where is God?"
And from within me, I heard a voice answer:
"Where is He? This is where--hanging here from this gallows." ...
That night, the soup tasted of corpses.
This quotation not only shows that Wiesel accepts this theodicy, but other Jews in Auschwitz do as well. This is one of the ultimate shows of doubt and anger for Wiesel. It almost subscribes to Nietzsche's "God is dead" idea; however, we must remember that it is an immediate reaction to the horrors the character is witnessing. The irony of the quote is that, by the end of the quotation, God still exists. He still may be "hanging here from this gallows," but He still exists. Here is yet another quote that proves this theodicy:
Never shall I forget that night, the first night in camp, which has turned my life into one long night, seven times cursed and seven times sealed....Never shall I forget those moments which murdered my God and my soul and turned my dreams to dust. Never shall I forget these things, even if I am condemned to live as long as God Himself. Never.
I find this quote incredibly ironic. There is admittance here that the Holocaust "murdered my God and my soul;" however, at the end of the quotation, Wiesel still admits the existence of God! This, very ironically, shows that God (even though He is not answering the tough questions) still exists everlastingly!
In conclusion, it is clear that this is the particular theodicy that Wiesel subscribes to (in my opinion) precisely because of Wiesel's "anguished questions" about God and what God allows.
2. Process Theodicy
In my opinion, Wiesel would reject this particular theodicy. Wiesel argues with God precisely because he thinks that God's power is NOT limited. This leads Weisel to (almost) reject God completely because he can't reconcile God being omnipotent with God's allowing the evils of the holocaust to exist.
3. Augustinian Theodicy
The key to this particular theodicy is that "all evils derive from the free will of finite beings-angels and humans." Again, I think that Wiesel would reject this particular theodicy. Why? Precisely because Wiesel DOES question God and has severe doubts. Someone who ascribes to this theodicy completely would not blame God at all for the holocaust, but point to the free will of the Nazis and how that free will (and, consequently the erroneous and evil decisions the Nazis made) was the ultimate reason behind the annihilation of so many Jews.
4. Person-making/Iranean Theodicy
An interesting concept, this particular theodicy hinges upon the immaturity of human beings. Because we are immature, we are not capable (yet) of making decisions worthy of the goodness of God. This would explain the Holocaust, for sure; however, in my opinion, Wiesel does not subscribe to this theodicy. If he did, there would be numerous passages of Wiesel blaming the Nazis fully and revealing their horrid immaturity. There would be no questioning of God at all.
Thus, we must say that the first theodicy you mention, that of a limited God, is what Wiesel subscribes to. This is the only theodicy that would describe Wiesel's "anguished questions" and yet still speak of God as existing. In my opinion, his faith is threatened, but not destroyed.