Many of his arguments draw on tensions that arise when believers assert that God is both (1) perfect and (2) all powerful.
Some of these argument are really just word games. For example, Korcz asks, can an all-powerful God create a rock he cannot lift? If he can't do both, he's not all powerful. But logically he can't do both. So he doesn't exist. Of course, that doesn't actually follow. This is just an argument against a certain understanding of all-powerful.
Similarly, Korcz asks, can an all-powerful, perfect God do something that is not perfect? If not, he's not all powerful. If so, he's not perfect! Gotcha! Again, this just begs the question, what do believers mean by "all-powerful." Korcz admits as much, saying if you qualify all-powerful, the contradiction disappears.
Believers have always recognized there are some limitations on God's power. Virtually all Christians believe life's purpose is to prove themselves worthy to return to god in heaven. Thus, the very purpose of life shows that God's powers are limited in some respect--either he cannot save everyone who is unworthy, or he cannot discern worthiness without performing a test.
These arguments are particularly unpersuasive to me because Mormons explicitly acknowledge limitations on God's power. Mormons recognize that God "cannot save [his people] in their sins" and that he is bound to keep his word when we [his people] do what he says.
The better arguments Korcz makes is the existence of evil. The argument from moral evils are pretty easily dismissed as God allowing his children to make moral (or immoral) choices. Since the purpose of life is to prove worthiness, it follows some of God's children will choose evil and impose costs on others. (Thus, the invocation of "libertarian" in the title to this post--although God does internalize externalities to the sinner in the afterlife).
The argument against God from natural evils is a hard one to answer, mostly because there are only possible answers. Kreeft's answer is that God allows these evils for reasons we either don't know or can't currently understand.
One possible answer is that, since our lives on earth are merely temporary and meant to provide us opportunities to learn and prove ourselves, natural evils are a necessary part of the test. It's a somewhat unsatisfactory answer, because we can't fully see the purpose of suffering natural evils or understand the ostensibly unjust distribution of the effects of natural evils. Still, it seems to me the best answer.
Another possibility is that God simply doesn't have control over some natural phenomenon, although I think conceding that probably limits the power of an all-powerful God too much.
Kreeft offers another interesting answer that natural evils were introduced by original sin--thus natural evils are linked to moral evils. An interesting argument, but one which makes me very uncomfortable, mostly because of loose-cannon preachers who blame victims of natural disasters for bringing the disasters upon themselves. I reject that particular invocation of the linkage, and so am tempted to reject the link all together.
So, in summary, I find most the arguments unconvincing, but concede natural evils do present a difficulty.