Let $A$ be a finite dimensional algebra (we can assume it is a connected quiver algebra). Call $A$ extfree in case for every indecomposable $A$-module $M$ we have $\operatorname{Ext}_A^i(M,M)=0$ for all $i>0$. One can show that $A$ being extfree implies that $A$ has finite global dimension and finite representation type. Thus it might be possible to give a classification of extfree algebras.
For example hereditary algebras of Dynkin type are extfree but there are also non-acyclic extfree algebras. The simplest example is the Nakayama algebra with Kupisch series [2,3,2,3].
Question 1: Does being extfree imply that $A$ is quasi-hereditary? (in all my examples this was true)
edit: Question 1 has a negative answer. A counterexample is the Nakayama algebra with Kupisch series [ 3, 4, 4, 3, 4, 4, 3, 4, 4 ].
Question 2: Is there a classification or another equivalent characterisation of extfree algebras? It would also be interesting to restrict this to a fixed quiver $Q$, for example $Q$ being a unique cycle so that $A$ is a Nakayama algebra.
Question 3: Is it more generally true that (noetherian) rings that are extfree have finite global dimension and are of finite representation-type?
