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.