Ext in symmetric algebras and group algebras

I think this example answers both questions.

Let $k$ have characteristic $3$, and let $G=C_3\times S_3$.

Then $kG$ has two simple modules, both one-dimensional, and for each simple module $S$, $\text{Ext}^1(S,S)$ is one-dimensional.

But if $M=kC_3$, with $S_3$ acting trivially, then $\text{Ext}^i(M,M)=0$ for $i=1,2$.

Suppose $A$ is commutative local (necessarily artinian) with the only simple $k\neq A$ then $\psi_k=1$, so your statement 1 will say that $Ext^1_A(M,M)=0$ implies $M$ is free. I stated it as a conjecture for complete intersections here (conjecture 9.1.3). Technically, it was stated as $Ext^1_A(M,N)=0$ implies $Ext^i_A(M,N)=0$ for all $i>0$, but when $M=N$ the latter condition is equivalent to $M$ being free.

One could also ask if $Ext^1_A(M,M)=0$ implies $M$ is free, still assuming that $A$ is Gorenstein. It was stated as a question (9.1.4) in the same survey.

As far as I know, both are open even for complete intersections unless $A$ is a hypersurface (which will be representation finite anyway).