Here is a counter-example to (c) for the semi-simple, but not simple, algebra $\mathfrak{so}_4$.
Projectives and Vermas are described in [Brüstle, Th.; König, S.; Mazorchuk, V. The coinvariant algebra and representation types of blocks of category $\scr O$. Bull. London Math. Soc. 33 (2001), no. 6, 669--681].
The self-dual projective with their notation looks like $$\begin{matrix} && d &&\\ & b && c\\ d && a && d\\ & c && b\\ && d &&\\ \end{matrix}$$
(Here $$\begin{matrix} d &&&&\\ & c &&&\\ && d &&\\ \end{matrix}$$ is a submodule, as can be seen from the quiver presentation given in that paper.)
If we quotient out $\Delta_c$, then the radical filtration looks like $$\begin{matrix} && d &&\\ & b && c\\ d && a && d\\ &&& b &\\ \end{matrix}$$ so the lengths of radical layers in the quotient are $1$ $2$ $3$ $1$. But the socle of this quotient has length $2$ (the left-most 'd' belongs to the socle), so the socle layers are different from the radical layers in this example.