Just some suggestions:
I am concerned you are talking about parameterization (as would be the natural thing in an engineering problem) rather than the intrinsic geometry of the curve itself (as a geometer should).
in your first case $v'(t) \propto v(t)$
in your second case $ \sum a_k(t)v^{(k)}(t) = 0$ for some time-dependent functions $a_k(t)$.
in your third case, it seems too much to have a third-order tangent everywhere but I'm not sure. who knows? You have certainly ruled it out.
I don't know how the everywhere existence of a particular catastrophe leads to a global restriction. There is a Gauss map from curves to projective space, or from surfaces to Grassmanian $\mathrm{Gr}(2,n)$ which encodes how the tangent plane changes as you move around the surface.