Consistent theory based on L and not(A->A) is a theorem
I'll assume that we're working in the context of classical propositional logic.
The first thing to point out is that $\lnot(A \rightarrow A)$ leads to a contradiction: ($A \land \lnot A$). Since propositional logic itself is consistent, it is impossible to derive a contradiction, so $\vdash \lnot(A \rightarrow A)$ is impossible.
So perhaps your intention was $T \vdash \lnot(A \rightarrow A)$. This can only happen if $T$ is itself inconsistent.
Given that $T$ is inconsistent, $T$ is vacuously complete (since an inconsistent theory can prove any formula) and decidable (for the same reason). As for soundness, it's a property of the logical system, not any particular theory like $T$. Propositional logic is sound.