First of all, I have a specific question. Suppose $M$ is an $m$-dimensional $C^k$-manifold, for $1 \leq k < \infty$. Is the tangent space to a point defined as the space of $C^k$ derivations on the germs of $C^k$ functions near that point? If so, is it $m$-dimensional? Bredon's book Topology and Geometry comments that only in the $C^\infty$ case can one prove that every derivation is given by a tangent vector to a curve. If so, this would suggest that (if indeed given this definition), the tangent space to a $C^k$-manifold would be bigger in the case $k < \infty$. Additionally, out of curiosity, would anybody have an example of a derivation that is not a tangent vector to a curve?
Secondly, it would seem to me that a fair share of the things I learned about smooth manifolds should fail or at least require more elaborate proofs in the $C^k$ case. We only used higher derivatives in proving Sard's theorem, but all the time we used the identification that the tangent space is given by tangent vectors to curves; the tubular neighborhood theorem comes to mind. What are the standard facts of smooth manifolds that do fail in the $C^k$ case?
Thirdly, are they really important? It seems a lot of books deal only with smooth manifolds, but a fair share also seem to deal with $C^k$-manifolds; Hirsch's Differential Topology deals with them all throughout, and Duistermaat's book on Lie groups defines them as $C^2$-manifolds. Should I, as a student of topology / geometry, be paying close attention to $C^k$-manifolds and the distinctions with the smooth case?