Q2: By Auslander-Reiten-Smalø, Theorem 1.9 page 65, says:

If $A$ is a finite dimensional elementary algebra over a field $k$ (that is, $A/\operatorname{rad}(A) \simeq k^n$ for some $n$), then $A\simeq kQ/I$ for some finite quiver $Q$ and some admissible ideal $I$ in $kQ$.

If $A\simeq kQ/I$ for a finite quiver $Q$ and an admissible ideal $I$ in $kQ$, the radical of $A$ is $\langle\textrm{arrows}\rangle/I$ and $A/\operatorname{rad}(A) \simeq kQ/\langle\textrm{arrows}\rangle \simeq k^{|Q_0|}$. Hence $A$ is an elementary algebra, and the above characterizes admissible quotients of path algebras.

This applied to $B=\operatorname{Ext}^*_A(M,M)$ gives that $M$ must be a basic module, that is, $M \simeq \oplus_{i=1}^n M_i$ with $M_i$ indecomposable and $M_i\not\simeq M_j$ for $i\neq j$. Furthermore, for $B$ to elementary $\operatorname{End}_A(M_i)/\operatorname{rad}\operatorname{End}_A(M_i)$ must be isomorphic to a fixed field $K$ for all $i$. This ensures that $B$ is a quiver algebras.

For Q1: Given the above, the software needs to be able compute the radical of $\operatorname{End}_A(M)$ and determine the structure of $\operatorname{End}_A(M)/\operatorname{rad}\operatorname{End}_A(M)$. For QPA this is to my knowledge only possible over finite fields. The command $\textrm{IsElementaryAlgebra}$ checks if a finite dimensional algebra over a finite field is elementary. To understand the structure of $\operatorname{End}_A(M)/\operatorname{rad}\operatorname{End}_A(M)$ is basically to decompose $M$ into indecomposable modules. The command $\textrm{DecomposeModule}(M)$ does this over finite fields. This gives a complete set of primitive idempotents of $\operatorname{End}_A(M)$, which correspond to the vertices in the quiver one wants to construct.

The arrows of degree $0$ is given by basis of $\operatorname{rad}\operatorname{End}_A(M)/\operatorname{rad}^2\operatorname{End}_A(M)$. The command $\textrm{EndOfModuleAsQuiverAlgebra( M )}$ would do this if the field $K$ is the same field $A$ is an algebra over.

Given that $\operatorname{rad}B = \operatorname{rad}(A) \oplus \operatorname{Ext}^{\geq 1}_A(M,M)$, we need to compute

$\operatorname{Ext}^n_A(M,M)/(\operatorname{rad}\operatorname{End}_A(M)\operatorname{Ext}^n_A(M,M) + \operatorname{Ext}^{n-1}_A(M,M)\operatorname{Ext}^1_A(M,M) + \cdots + \operatorname{Ext}^1_A(M,M)\operatorname{Ext}^{n-1}_A(M,M) + \operatorname{Ext}^n_A(M,M)\operatorname{rad}\operatorname{End}(M))$

to find the arrows of degree $n$. The command $\textrm{ExtAlgebraGenerators}(M, n)$ would find $\operatorname{Ext}^n_A(M,M)$ modulo everything (among other things) except for the two outer terms (so the documentation on this command is misleading as it is only correct if the degree zero part is a semisimple ring). However all of this is possible to do in QPA, but there is no readily available commands to preform these tasks. These necessary commands are something we want to implement in QPA version 1 or version 2.

I hope that these are useful comments.

Best regards, The QPA-team.