Language of particular CFG
First, let us notice the following:
$$
L(S) = \{ xyz \mid x,z \in \{0,1\}^*, |x|=|z|, y \in L(A) \} \\
L(A) = 0L(B)1 \cup 1L(B)0 \\
L(B) = (0+1)^*
$$
Putting everything together, we get
$$
L(G) = \{ x\sigma y \tau z \mid x,y,z \in \Sigma^*, \sigma,\tau \in \Sigma, |x|=|z|, \sigma \neq \tau \}
$$
In other words, a word $w$ is in the language if there exists $i \leq |w|/2$ such that the $i$th symbol from the left is different from the $i$th symbol from the right. The condition $i \leq |w|/2$ can be eliminated, and we get that for some $i$, the $i$th symbol of $w$ is different from the $i$th symbol of $w^R$. In other words, $w \neq w^R$.

So your grammar generates the language of non-palindromes.