To be specific, I'll focus on the second question of the title, "Is PA consistent? do we know it?"
As noted by Noah Snyder on the meta thread, this question itself already uses "philosophical" language like "know". So I think it may be viewed it as
$\bullet$ a question in mathematics, if you accept a specific philosophical position such a form of platonism, formalism, constructivism, or ultrafinitism; or
$\bullet$ a question in philosophy of mathematics, if you remain ignostic regarding the meaning of "know".
So as to stay on-topic, I will only consider the first possibility, and only those four positions that it explicitly lists. In fact, for a constructivist's view I absolutely support what Emerton said (only I'm not sure about the border between what Voevodsky actually claimed in his talk vs. what he might have meant), and for a formalist's view I agree completely with Todd Trimble's answer. (Indeed, before these two answers appeared I had felt something of those kinds was badly missing from this discussion.) Regarding an ultrafinitist's view, I think Mirco Manucci has a point, and I'll incidentally elaborate on another important point made by Qiaochu Yuan on the meta thread.
So I will now pretend that I'm a platonist (which is what I usually do when attacking some problem - but certainly not when writing up a proof) and in doing so I'll try to argue that Timothy Chow's first answer is simply wrong. ("Wrong" in platonist's absolute, undefinable sense.)
The problem with Timothy's argument is that it unfairly exploits our subconscious reading of some words in a mathematical text.
When I'm stating a theorem, in a paper or in a talk, I almost never put it like "Theorem (ZFC)." or "Assume ZFC. Then ..." because my area of geometric topology is (or at least is commonly thought to be) very far from foundations, and such pronouncements, commonly seen as tautological, would be very distracting. But I do mean it. (Not that I like ZFC too much, especially so with the uncountable "C" which strikes me as really awful in the context of ZF. But I recognize that socially and in terms of the existing literature to refer to, I don't have much choice - so all my results are, alas, meant to be in ZFC by default.) I also don't start papers and talks by saying that everything will be in ZFC, for the same reason that this information is (as long as my field is concerned) obvious and distracting, and hence is likely to be a repelling factor; and moreover because of my personal negative emotions towards ZFC. I did, in fact, on one occasion stated a "theorem (ZFC)" in a talk, but only to emphasize that I'm not assuming CH or PFA like some previous authors did.
I'm assuming that other authors may have somewhat similar considerations, so in papers, books and talks in my area, I'm reading every "theorem" as a "theorem (ZFC)", or at least as what could have been meant by the author to be a theorem in ZFC. There seem to exist people who don't read theorems in this way (I'm thinking, in particular, of applied mathematics and mathematical physics; of Vladimir Arnold; and of experimental mathematics and computer science) but certainly N. Bourbaki and his faithful readers do read them in this way; I also suspect some countries including France and Poland to have more of this tradition than some other ones. The same of course applies to lemmas, problems and conjectures.
Now the situation is different in foundations. You certainly don't read "Problem. Is PA consistent?" as "Problem (ZFC). Is PA consistent?" - at least if you have ever heard of Hilbert's, Goedel's and Gentzen's work on this subject. In set theory, authors usually make it clear what formal system is assumed in their book, chapter or theorem. In other subjects such as proof theory and model theory I understand that the situation is rather complex, but it seems that the prevalent convention in some areas where PA or the second-order arithmetic are more relevant than ZFC is that by default, a "Theorem" could read as "Theorem (no hypotheses whatsoever)." What exactly "no hypotheses whatsoever" means is a separate question, but for the moment I'd like to note that the linguistic/pshychological issue of the clash of conventions for reading words like "theorem", "problem" and "proof" is not mentioned in Timothy's first answer, yet is central to its understanding. For this reason I dismiss his argument as pure sophistry.
This doesn't yet address his conclusion - that Con(PA) is true (in the sense of Plato) because we know (in the sense of Plato) a model of PA. (By the way, this purported knowledge usually turns out to be an implicit hypothesis in the "no hypotheses whatsoever" reading.) But how could we possibly know that what we usually refer to as "1,2,3,..." (I will abbreviate this as $\Bbb N$) is indeed a model of PA? (Note that "1,2,3,..." is only a name, and not a model itself, due to the presence of the undefined/circularly defined symbol "...".) I see only 3 possibilities:
by virtue of a religious belief;
we could know it from experience, by having a physical model of PA;
we could know it from the mind, by demonstrating logically that $\Bbb N$ must be a model of PA.
In connection with the off-topic possibility (1), which I'm not denying, let me only mention some sources: (a) Poincare has devoted quite a few interesting pages to argue
for his view that the axiom of induction is a synthetic a priori judgement,
(b) Goedel, and in more detail Roger Penrose used the hypothesis that $\Bbb N$ is a model of PA to argue rather convincingly for certain philosophical propositions related to religion.
The off-topic possibility (2) would have strong consequences for physics, which have not been established yet: given a physical model of PA, apparently either the "physical universe" (the past light cone) is a non-compact 3-manifold; or there can be an infinite amount information within its bounded region, which would contradict the holographic principle (which holds in some flavors of string theory) and also some rivals of string theory such as loop quantum gravity (which involves a quantized, rather than Euclidean, space-time).
Finally, the possibility (3). Parikh and Sazonov have shown (assuming our knowledge of a model of PA) that there exist "truncated" versions of PA which are contradictory due to the truncation, yet the shortest proof of contradiction is too long to fit within the theory; thus the theory doesn't "see" itself as being contradictory. Now imagine a finite computer $X$ on which a Peano-Sazonov arithmetic is implemented. If we happen to have a model of PA, or a bigger computer $Y$ (without tricks) at hand, we could be able to use $Y$ to verify that $X$ does not emulate PA correctly, and so $X$ could be finite. But without $Y$, I don't see how to do it. So it looks like $X$ must seem infinite to its user (unless the user's mind is bigger than $X$).
Could our $\Bbb N$ be just a kind of the computer $X$? That is, could God have faked Sazonov integers for us, and kept Peano integers for himself? I don't see any reason why this couldn't have happened. Please correct me if I'm wrong. (I can't resist recalling that while most people are aware nowadays that the Earth is not flat, the notion of $3$-manifolds other than $\Bbb R^3$ still does not occur to many people outside academia in connection with the physical universe.)
The conclusion is that we don't know that $\Bbb N$ is a model of PA; it is an open problem (in platonist's sense, no hypotheses whatsoever, in particular no assumptions regarding our knowledge of a model of PA). Hence there's also no known reason why Con(PA) couldn't be an open problem (in the same sense).
