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.