CH is orthogonal to ZF. CH is orthogonal to ZFC.
If ZFC is inconsistent, then ZF is also inconsistent.
AC is orthogonal to CH.
That doesn't actually answer Douglas's statement that the continuum hypothesis is orthogonal to everything people care about if one assumes choice. In fact Doug's statement is more or less correct. See in particular discussion here. In particular, ZF + CH implies choice for sets of real numbers, which is what we care about for most practical purposes.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.