Certainly $B=C_2=0$ for CSS codes. But for the other two.
Argument reductio ad absurdum
A CSS code is made out of two classical codes, with parity check matrices $H_X, H_Z$, and these matrices are the submatrices of the quantum code's stabilizer generator matrix.
Classical codes can also be transformed into equivalent codes by row operations on the parity-check matrices.
Let's imagine if $H_i$ is a $k_i\times n$ matrix for $i=X,Z$, and represents a $n$ bit code. Now, if $H_i$ could be simplified further so that entire columns could be made zero, then it is actually a code on fewer than $n$ bits! That is clearly not possible.
Argument impossible algorithm
We have the matrix
$$
H = \left(\begin{array}{ccc|ccc}
I & A_1 & A_2 & 0 & 0 & 0 \\
0 & 0 & 0 & D & I & E_2
\end{array}\right).
$$
How do we intend to turn any of the submatrices $0$ with row operations? Take $D$ for instance. There are no $1$ in any row above it. So we can't do any operations that remove all $1$ from every column of $D$. We will always be left with at least one $1$ in each column no matter what row operations we do.