The three questions laid out are the same anyone can ask of anything, What is it? How can we know it? and How can we express it or communicate it? The Greek word Logos (λόγος) is a powerful word to express all these ideas together, intelligible being/reality, human knowledge/wisdom/reason/science, and language/communication. The ancients/premoderns focused on the first part, metaphysics, and moderns, starting with Descartes, were more concerned with the second question, epistemology.
After Thomas' five ways to God he summarizes the way he's going to approach these questions about knowing God (as quoted in the lecture notes, the brackets are Prof Kreeft's interjection):
"When the existence of a thing has been ascertained, there remains the further question of the manner of its existence, in order that we may know its essence. [“Essence” means “manner of existence” for Aquinas.] Now . . . we cannot know what God is but rather what He is not . . . therefore we must consider, first, how He is not, second, how He is known by us, and third, how He is named."Though Thomas believes that faith, divine revelation, and religious experience are all valid ways of knowing God, this discussion focuses on philosophical/epistemological knowledge by natural reason. Thomas seeks to show that there are (at least) four metaphysical principles that answer how man can know God: creation, cosmic hierarchy, the analogy of being, and human reason as participating in divine reason. If any one of these weren't true then man couldn't know God by natural reason.
Creation, this doesn't tell us much about God, but it does tell us some. As Thomas' five ways show that God created the universe, now the fact that He did so tells us a bit about Him.
From effects not proportionate to the cause no perfect knowledge of the cause can be obtained. Yet from every effect the existence of its cause can be clearly demonstrated, and so we can demonstrate the existence of God from His effects even though from them we cannot know God as He is in His essence.So we can know that God is a God of cause by the effects we see, though that doesn't tell us much about Him. Not a complete rationalist/gnostic or agnostic, but a little more agnostic.
Cosmic Hierarchy, implied in creation is a sort of hierarchy that God is the top of the cosmic ladder of existence and mankind is somewhere on this ladder below God. Incidentally, Thomas didn't think you could prove the existence of angels, but he thought it would certainly make sense that there be at least one level of existence between God and mankind, just as there are different levels between mankind and a slug.
Analogy of being, this goes along the same lines as cosmic hierarchy that along the cosmic ladder of existence there are bit of analogy. In the case of life, there is an analogy in that there are lower forms of life but they're still life and there are higher forms of life. The highest in divine life, not that God's existence mirrors ours but that we mirror God's existence because we get our existence from Him. This same chain of analogy is in logos/orderliness. Even the simplest pieces of matter are orderly. In fact, to me the more order we see at these "lower" levels is a clear indication of design. So, along the rungs of this ladder of analogy the higher levels have more order and reflect God's orderliness (and other characteristics) better. Until you get to the top of the ladder, God, from which all orderliness and design comes.
Divine reason, this is not some mystical experience, it's more akin to the idea that all logic and reason finds its source in God. This part of the lecture actually reminds me of presuppositional apologetics (as I've discussed with +James Hooks a few times). To medieval logicians there were three basic acts to the mind conception, judgement, and reasoning from judgements. First, one must recognize concepts, "man," "apple," "animal," "is," "not" etc. Then one judges from these simple concepts, "man is an animal," "apples are not animals." Then finally, the reasonable moving from premises to conclusion(s) "therefore man is not an apple." Thomas is referring to all three of these acts in saying that man's intelligence is a dim reflection of God's divine intelligence. God doesn't have to judge and reason, He just knows everything all the time, but mankind as a small reflection of that, can only do bit by bit. As a dim reflection of God, we share in His divine reason/intelligence as participating in His intelligence. One of the main presuppositions in this part of the argument is that God is rational/mind, not just a blind force. There are many assumptions in Thomas' philosophy as it's more like a mapping of a complex universe than the dry-step-by-step-deduction-only-philosophy of the moderns.
Now, in light of these four assumptions we should try to understand what Thomas actually says about knowledge of God. Remember that he's closer to agnosticism than rationalism (gnosticism). Prof
Kreeft has this to say about the articles immediately preceding the ways to God in the Summa, "...God’s existence is not self-evident to us, though it is self-evident in itself. So it has to be made evident to us... God is not directly and innately known—our mind is not that strong—but His existence can be made known by reason, can be demonstrated—our mind is not too weak for that." (Quote from the lecture notes, emphasis mine.) This is from the next part of the Summa about being able to know about God.
Since the ultimate beatitude of man consists in the use of his highest power, which is the operation of the mind, if we suppose that the created intellect can never see God, it would either never attain to beatitude or its beatitude would consist in something else besides God. This opinion is against reason, for there resides in every man a natural desire to know the cause of any effect which he sees, and thus arises wonder in man. But if the intellect of the rational creature could not reach so far as to the first cause of things, this natural desire would remain void. But no natural desire is in vain.So, we may not be able to fully know God in this life but we will attain sure knowledge of God in the next. Thomas uses this statement many times, that “no natural desire is in vain.” Seems reasonable to assume that the desire for a more perfect understanding of something and that that desire can be fulfilled though perhaps not fully in this life.
There are two levels of understanding here that we must distinguish. Comprehension and apprehension, we as created beings will never be able to surround in knowledge, God. But, we will someday, be able to apprehend God. Just how far can we get? Thomas give four things we can know: He exists, He is the cause of creation, that He has a certain deducible attributes, and what He is not: He is not a creature and not at the same level as creatures. In His being, He is infinite, unlimited being in itself, while creatures only have being in a limited way sourced from God.
According to Thomas outside of analogical knowledge our knowledge of God is only negative. The reason for this is the fact that we are created, finite creatures and cannot know by experience things that apply only to God, like infinitude. We can see the art that only resembles the Artist but we cannot know all that is in the Artist.
This next bit I really liked so I'm just going to let Prof Kreeft speak for himself (quote taken from the lecture study guide):
"I suspect that most of the time, belief or unbelief in God’s existence depends on understanding the meaning of the term “God.” I never met an atheist who I thought fully understood what an intelligent theologian like Aquinas meant by “God.” After talking for a while with an intelligent atheist I always find myself agreeing with him in denying the God he denies; only I claim that’s not the God Aquinas is talking about. For instance, a God who moves around in time and changes, and therefore gets either better or worse in some way, or a God that’s timeless and changeless by doing nothing. Both of these concepts of God are imperfect. Aquinas says God’s perfection is pure act, pure actuality, and pure activity of knowing and loving. He doesn’t change, He doesn’t learn truth because He is all truth, and He doesn’t fall in love for the same reason water doesn’t get wet: because He is love. "I've seen this in many arguments against God, including my previous series from Prof McGinn.
This next question is also eloquently answered by Prof Kreeft, it's the question commonly brought up even as recent as this post on Google+. The answer isn't really all that complicated! It's a linguistic problem, that goes something like this: God has infinite power so asking this question is like asking if a rock could exist that infinite power couldn't pick up. So, the easy answer is no, but not that God isn't all-powerful, it's a misunderstanding of infinite power. A similar answer can be given to many of the questions concerning God. Who created God? That's like asking, what existed/happened before infinity past? That's a contradiction. My personal view is that we shouldn't limit God to a logical anwer, but within our limited perspective, these questions don't make any sense.