Let $M$ be a real $C^k$-manifold, $\mathscr{H}_x$ be the $\mathbb{R}$-algebra of germs of $C^k$-differentiable functions at $x \in M$. It is easy to show that $T_x M$ can be embedded into the space of derivations of the form $v: \mathscr{H}_x \to \mathbb{R}$. In his book on Lie groups and Lie algebras Serre proves that in case of analytic manifolds this embedding is indeed an isomorphism of vector spaces.
I was told that this is also true in case of $C^\infty$, but there are derivations which are not tangent vectors when $k < \infty$, and the definition of the tangent space (cf. here) is quite different, and these definitions are only shown to be equivalent when $k = \infty$.
So my questions are:
1) What is the correct way of defining the tangent space, after all? What breaks down if we try to use maximal ideals when $k < \infty$? An (obscured by adding $\mathbb{R}$) explanation is given here.
2) Can you please give me an explicit example of a derivation which is not a tangent vector? There is a proof of existence here, but no explicit example.
Also, a softer question: are $C^k$-manifolds ($k < \infty$) important in differential geometry or differential topology? E.g. in Dubrovin-Novikov-Fomenko's textbook the Sard's theorem is proved for $C^\infty$ and there was no mention of the case when $k < \infty$, this leads me to believe that $C^\infty$ and $C^\omega$ are the only important cases in differential topology. Is this really so?