Wayne P. Pomerleau writes in the Internet Encyclopedia of Philosophy article already referenced in the question:
Kant deals with the problem of evil more impressively in his “On the Miscarriage of All Philosophical Trials in Theodicy” (1791). He analyzes possible attempts at theodicy into three approaches: (a) it can argue that what we consider evil actually is not, so that there is really no conflict; (b) it can argue that the conflict between evil and God is naturally necessitated; and (c) it can argue that evil, though contingent, is the result of someone other than God. Kant’s own earlier work attempted to combine the second and third strategies; but here he concludes that all of these approaches must fail. More specifically, attempts to show that there is no pernicious conflict between moral evil and God’s holiness, between the physical evils of pain and suffering and God’s goodness, and, finally, between the disproportion of happiness and misery to virtue and vice and God’s justice, all fail using all three approaches. Thus Kant’s considered conclusion is negative: the doubts that are legitimately raised by the evil in our world can neither be conclusively answered nor conclusively refute God’s infinite moral wisdom.
This would answer the first question:
- Did Kant discuss theodicy at all?
He did discuss theodicy in “On the Miscarriage of All Philosophical Trials in Theodicy” (1791).
One might be able to answer the second question as well:
- What possible explanations did Kant supply for good things happening to bad people and bad things happening to good people?
Based on the above quote he did not provide an explanation, however, he found "no pernicious conflict between moral evil and God’s holiness". He leaves this to faith.
Lawrence Pasternack and Philip Rossi give an explanation for Kant's reliance on faith:
So, while Kant does deny the possibility of religious knowledge (as well as opinion), he considers this denial necessary to safeguard faith, as the proper mode of religious assent. One must, therefore, understand the negative elements in his philosophy of religion, such as his infamous objections to the traditional proofs for God's existence, in this context.
One needs faith for "religious assent".
Pasternack, Lawrence and Rossi, Philip, "Kant's Philosophy of Religion", The Stanford Encyclopedia of Philosophy (Fall 2014 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/fall2014/entries/kant-religion/.
Pomerleau, W. P. "Emmanuel Kant: Philosophy of Religion" Internet Encyclopedia of Philosophy https://www.iep.utm.edu/kant-rel/