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Is there a high-concept explanation for why characteristic 2 is special?
There are so many results on primes that either fail for $p=2$ or are not known to be true for $p=2.$ Can anyone give some kind of intuition as to why this is the case?
My take on this issue is that p=2 isn't really strange---all small primes are strange, it's just that the smaller you are, the earlier you become troublesome. Look at recent R=T results in the theory of automorphic representations. Nowadays people can prove these sorts of things for $n$-dimensional representations, but they need to assume $p>n+1$ or some such thing. The thing about $p=2$ is that it's so small that it's already causing problems when one is considering $GL(1)$, which is an abelian situation. Now abelian situations are so much easier to understand than the general situation that they are more prevalent in the literature. For example things like quadratic reciprocity can be viewed of as some consequence of class field theory, which is really 1-dimensional representations of Galois groups, and already $p=2$ is causing a problem. Similarly Fontaine's results on commutative group schemes of $p$-power order runs into some trouble when $p=2$ (his basic linear algebra data doesn't give you an equivalence between finite flat group schemes over ${\mathbf Z}_p$ and "easy semilinear algebra" when $p=2$) and again it's because 2 is just too small. But as people formulate higher-dimensional analogues of these things, they will no doubt have to rule out more primes. So it's not that 2 is behaving badly, it's just that from 2's point of view the theory is more advanced, so you have to deal with more special cases.
Another reason that 2 is a strange prime is that for a prime $p$, the divided powers $\frac{p^n}{n!}$ tend to zero $p$-adically unless $p=2$. This makes many things in the theory of crystalline cohomology, $p$-divisible groups and integral $p$-adic Hodge theory more subtle (and in some cases just false) when $p=2$.
I guess the standard answer is that, in the field of $p$-the roots of unity, the prime $p$ behaves differently. In the rationals, which contain the 2nd roots of unity, the prime $2$ is expected to show a strange behaviour. In my experience, the prime $2$ behaves half as odd if it is replaced by $4$.