Let $ A$ be a finite dimensional (connected) quiver algebra. Let $ T(A)$ denote the full subcategory of coherent functors from $ modA$ to $ Ab$ that vanish on projective objects. $ T(A)$ is equivalent to the module category of the stable Auslander algebra of $ A$ when $ A$ is representationfinite.
The indecomposable projective objects of $ T(A)$ are $ \underline{Hom_A}(,X)$ for any indecomposable nonprojective object $ X$ and this is noninjective if and only if $ X$ has codominant dimension equal to zero. So assume in the following that $ X$ is nonprojective with codominant dimension zero (meaning that the projective cover of $ X$ is not injective).
Auslander and Reiten showed (see for example their article “stable equivalence of artin algebras”) that a injective coresolution of $ \underline{Hom_A}(,X)$ is given as follows when $ 0 \rightarrow \Omega^1(X) \rightarrow P \rightarrow X \rightarrow 0$ determines the projective cover of $ X$ :
$ 0 \rightarrow \underline{Hom_A}(,X) \rightarrow Ext_A^1(,\Omega^1(X)) \rightarrow Ext_A^1(,P) \rightarrow Ext_A^1(,X) \rightarrow Ext_A^2(,\Omega^1(X)) \rightarrow Ext_A^2(,P) \rightarrow Ext_A^2(,X) \rightarrow Ext_A^3(,\Omega^1(X)) ….$
Questions:

In general this injective coresolution will not be minimal. Is it known what the minimal injective coresolution looks like?

When is the above injective coreslution minimal (or at least minimal until the first term is zero)? I suppose that this is for example the case when $ P$ and $ \Omega^1(X)$ are indecomposable.

Is there are formula for calculation the injective dimension of $ \underline{Hom_A}(,X)$ or at least when it is finite?

In case every indecomposable summand of $ X, P$ and $ \Omega^1(X)$ has infinite injective dimension, does also $ \underline{Hom_A}(,X)$ has infinite injective dimension?