Classifying Space of a Group Extension

Yes. The principal bundles are the same and your guess that $BA$ is an abelian group is exactly right. A good reference for this story, and of Segal's result that David Roberts quotes, is Segal's paper:

G. Segal. Cohomology of topological groups, Symposia Mathematica IV (1970), 377- 387.

The functors $E$ and $B$ can be described in two steps. First you form a simplicial topological space, and then you realize this space. It is easy to see directly that $EG$ is always a group and that there is an inclusion $G \rightarrow EG$, which induces the action. The quotient is $BG$. Under suitable conditions, for example if $G$ is locally contractible (which includes the discrete case), the map $EG\rightarrow BG$ will admit local sections and so $EG$ will be a $G$-principal bundle over $BG$. This is proven in the appendix of Segal's paper, above. There are other conditions (well pointedness) which will do a similar thing.

The inclusion of $G$ into $EG$ is a normal subgroup precisely when $G$ is abelian, and so in this case $BG$ is again an abelian group.

I believe your question was implicitly in the discrete setting, but the non-discrete setting is relevant and is the subject of Segal's paper. Roughly here is the answer: Given an abelian (topological) group $H$, the $BH$-princical bundles over a space $X$ are classified by the homotopy classes of maps $[X, BBH]$. When $H$ is discrete, $BBH = K(H,2)$. If $X = K(G,1)$ for a discrete group $G$, these correspond to (central) group extensions:

$$H \rightarrow E \rightarrow G$$

If $G$ has topology, then the group extensions can be more interesting. For example there can be non-trivial group extensions which are trivial as principal bundles. Easy example exist when H is a contractible group. However Segal developed a cohomology theory which classifies all these extensions. That is the subject of his paper.

Your picture is essentially correct, except you need to specify that the maps take basepoints to basepoints (i.e., they are pointed maps).

BG is given as the homotopy fiber of a pointed map from B(G/H) to BBH. Applying the based loop space functor $\Omega$ to the pointed maps yields the sequence of group homomorphisms. This does not require G to be abelian. There is a subclass of abelian central extensions, which are those that can be delooped again to maps from BB(G/H) to BBBH.

In general, group homomorphisms can be delooped once to maps of pointed spaces, while abelian group homomorphisms can be delooped arbitrarily many times, to infinite loop maps of infinite loop spaces. There is an equivalence between grouplike homotopy commutative spaces and double loop spaces, but since G is discrete, the double loop space property is strictified for free.

Edit: A standard reference for this material is the second chapter in Adams, Infinite Loop Spaces.

@There is an equivalence between grouplike homotopy commutative spaces and double loop spaces

No, grouplike homotopy commutative is not enough. Double loop spaces have much more in thee way of higher homotopies, even if the space were to have a strictly associative homotopy commutative structure. That was one of the motivations for operads. Also see JF Adams: 10 types of H-spaces.