Another (related) property that fails is that inverses stop being useful for cancellation. Inverses still exist, for every p there's a q with pq = qp = 1, but if you've got an equation ap = b you can't cancel to get a = bq, because we don't have associativity. The left hand side (ap)q doesn't equal a(pq), so you can't reduce it to a.
Of course associativity doesn't hold in the octonions either, but it holds just enough for cancellation to work.
Yes, and this is a bit obvious, but reals, complex numbers, split complex numbers, quaternions, octonions, sedenions, can all be represented as matrices of the appropriate form.
That's at most sort-of-true. It's not possible to represent octonions by matrices of numbers in such a way that multiplication of matrices corresponds to multiplication of octonions, because matrix multiplication is associative and octonion multiplication isn't.
Of course associativity doesn't hold in the octonions either, but it holds just enough for cancellation to work.