Here is a counter-example to (c) for the semi-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
$$\begin{matrix}
&& d &&\\
& b && c\\
d && a && d\\
& b && c\\
&& d &&\\
\end{matrix}$$

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 lenght $2$ (the right-most 'd' belongs to the socle), so the socle layers are different from the radical layers in this example.