His answers are kind of longish, so allow me to summarize:
1) Why is there anything?
God made it. (The Roman Catholic one, because Egnor is a Roman Catholic, so he knows that it was that one that did it. Somehow.)
2) What caused the Universe?
God caused it.
3) Why is there regularity (Law) in nature?
Originates in God.
4) Of the Four Causes in nature proposed by Aristotle (material, formal, efficient, and final), which of them are real? Do final causes exist?
All four causes are real. Final causes exist.
[Final cause is the end, goal or purpose of a thing (the final cause of a rubber ball is to provide a bouncy toy). Egnor thus espouses the view so heavily criticized (though not universally so) by evolutionary biologists that every organ/trait has one purpose, and that purpose is why the organ/trait is there. No wonder true vestigial traits (those of no function at this time) is such a big problem for Egnor and his fellow Id proponents.]
5) Why do we have subjective experience, and not merely objective existence?
Because God gave us a soul.
6) Why is the human mind intentional, in the technical philosophical sense of aboutness, which is the referral to something besides itself? How can mental states be about something?
Apparently, that was an easy one: Intentionality is easily explained; my thought that is instantiated in my brain state refers to an apple because the form of the apple is grasped by- is actually taken into- my mind.
But seriously, I cannot possibly summarize his answer.
7) Does Moral Law exist in itself, or is it an artifact of nature (natural selection, etc.)
Moral law is objective, and made by God.
8) Why is there evil?
Because Adam and Eve sinned by eating a fruit that made them blush about their genitals and God became (and continues to be) really, really angry. In other words, God allows/created evil.
★ ★ ★
Egnor also refers to current evolutionary theory (and the rest of science) as incapable of explaining any of these things. For example:
7) The assertion that Moral Law is subjective or is a byproduct of evolution is incoherent at best, and has horrendous implications for mankind.
Sigh! Moral sentiments are rather easily explained by evolutionary theory. And the second statement is clearly false, because there are many atheists worldwide, and they are not the problem. (And despots were religious and irreligious, both.)
8) If mankind evolved by natural selection, we wouldn't even perceive the death of unrelated others as evil. It would be a real win- more offspring for me!
Yes we would. We do. Again, empathy can be explained by evolution, too. Plenty of good books on that, such as Evolutionary Origins of Morality or The Evolution of Morality.