A consequence of your definition is that your local fractional derivative vanishes for all Lipschitz continuous functions, and even on Hölder continuous functions with exponent $>\alpha$. Huge kernel !

Yes: the method devised by Newton solves the equation $\Phi(X)=0$ at simple zeroes. Somehow these zeroes are stable by small perturbation and are much easier to find than the zeroes at which the differential is vanishing.

This business has to do with the $J^t=\exp{itD_x\cdot D_\xi}$: the adjoint of $op_t(a)=op(J^t a)$ is $op_{1-t}(\overline a)$, so somehow the only way to have full stability with respect to taking adjoints is indeed the Weyl choice. Of course, you have always asymptotic stability for good classes of symbols, say up to $h^\infty$ in the semi-classical case, but it is not very satisfactory at the algebraic level.

@Denis Serre Are you sure of this ? After all constant coefficients PDE are described by a polynomial with $n$ variables, a very simple type of analytic function. Why the preparation theorem, usually devised to dealing with smooth non-analytic functions satisfying a finite type assumption could be useful in that context ?

@Elena You have plenty of Eulerian results (I mean for the vector field PDE) for that type of vector fields. In particular a MR search on Ambrosio as author and keyword "nearly incompressible" or "$BV$ vector field" will provide plenty of references.