Synthetic a priori knowledge of mathematics and geometry is the starting point for Kant's reconciliation of the two schools carried out in Critique of Pure Reason.
Kant did not advocate taking anything on faith in the sense you are using the term.
What Kant did was point out that all empirical knowledge is mediated by the conditions of human experience (i.e. space and time), and that we should not confuse empirical facts with the ding an sich (thing in itself).
Kant's most famous quote is along the lines of "So act that your principle of action might safely be made a law for the whole world."
It usually leads to people justifying situational ethics by mentioning Nazis.
Exactly right. The sense in which the parent uses the term "faith" is a fairly modern invention. It's only over the last hundred years or so that "faith" has been thought of as the opposite of reason.
In ancient Jewish and Christian writings, "faith" is typically used to mean "acting on something you know to be true", especially in the face of difficult circumstances. It's typically contrasted with forgetfulness or fickleness. In this usage, "faith" is the triumph of reason, memory/history, and will over the emotions and difficulties that come with temporary adversity.
In Kant's usage, "faith" is sort of an extension of reason that also includes elements of willpower and morality. It's pretty close to the ancient definition, though it also played an important part in the shift from the ancient definition to the modern one.
You're talking about ancient religious writings. Religious people have a marked tendency of incorrectly conflating faith and reason. Clearly, whatever they meant by "faith," it had a strong mystical (i.e. otherworldy, not based in this reality) slant.
The exact same point goes for Kant. You call his "reason" the following: "reason that also includes elements of willpower and morality" (which does hint at Nazi ideology). I call Kant's "reason" the following: a flawed, mistaken, weakoned account of reason, plus some mystical aspects to compensate.
A small but vocal subset of religious people within the last 100 years (particularly fundamentalist evangelicals) have a marked tendency of treating faith as a strongly mystical thing which is opposed to, and better than, reason. But within most of the rest of Judeo-Christian history and tradition (including most modern Jewish and Christian intellectuals), faith has been thought of in the way I described it: the triumph of reason and experience and willpower, in opposition to fickleness and forgetfulness and emotion.
The only ancient mysticism I know of surrounding Jewish or Christian ideas of faith is the idea that a person might have a supernaturally strong will, and therefore may be able to continue acting according to reason through remarkably dire circumstances. The concept of faith itself, within those traditions, is not mystical or supernatural.
Kant's morality has mystical aspects, and therefore, by association, so does his faith -- as I said, it was a step toward the modern definition (and it may very well be flawed.) Ayn Rand's definition is particularly 20th-century influenced: "blind acceptance of a certain ideational content, acceptance induced by feeling in the absence of evidence or proof." You will not find this mystical, blind conception of faith prior to about a hundred years ago; it is not what was meant by Kant, Augustine, Paul, or Moses.
Kant used the analytic/synthetic dichotomy as part of his attempt to "destroy" reason. No, this isn't the same as a religionist's call to faith, but it has the same implications.
I am familiar with the categorical imperative, but it's not directly related to the discussion at hand. Yes, Kant is also very famous for that, and I also disagree with that (as does most everyone, probably even yourself :-) ).
The author is conflating two philosophical schools of thought, Rationalism and Empiricism, with two other concepts (with a complex relationship), reason and science.
Point 1: The author is defining reason incorrectly. He states that reason is "internally generated," that it starts with "your internal sense for what is right and pure." The correct definition (taken from a dictionary) is: the mental powers concerned with forming conclusions, judgments, or inferences. The difference is that the author claims that reason is basically internal/introspection, whereas in actuallity, reasoning requires both internal mental processes (such as recalling concepts you've formed previously) and extrospection (observing the world around you, learning from it, etc.). The Rationalist school of philosophical thinkers (e.g. Descartes) argued that that extrospection is fundamentally flawed or not trustworthy, so the definition the author uses is actually, basically, the rationalists' position. (Later philosophers like Kant picked up with this and ran with it, claiming that since reason is faulty, you must resort to faith. Indeed, the rationalists were typically highly religious, although basically everyone was at that time.) I, personally, disagree very strongly with that position, and I suspect that many HN readers also disagree with it. So be careful not to accidentally "accept" this position as a valid definition.
Point 2: Empiricism is a philosophical school that was formed in reaction to the Rationalists. Empiricist philosophers pointed out that the Rationalists were incapable of achieving true knowledge about the world by rejecting perception, and insisted that we just go by experimentation. However, they accepted some of the Rationalist viewpoints about the way reason works, and thus were very skeptical of reasoning, prefering "hard data" and the like to explain things. (This is ironic because validating a philosophical claim would require reason, so you can't use philosophy to reject reason without contradicting yourself, but I digress.)
Point 3: The actual relationship between science and reason is not "science vs. reason" or "science OR reason", as the author states. It's "you need to use reason to do science." Science is a kind of technical study of the way the world works that uses certain techniques; that's all science means. There's things you can do using reason that aren't technical enough to be called science (like paying your taxes or thinking about philosophy). But there's no science you can do without using reason.
Conclusion: The claim that reason and reality are fundamentally divided is the root premise of the author's blog post. This was Kant's position. Kant's main goal was to state that reason isn't really useful, so we all have to go on faith; he was trying to uphold "God, faith, and immortality" (to cite his most famous quotation). That gives me the creeps. I strongly disagree with Kant.