Given a Dynkin quiver $Q$ and a field $K$.

Question 1: For which such $Q$ are there only finitely many indecomposable representations over the dual numbers $K[x]/(x^2)$?

Note that those representations are exactly those of $KQ \otimes_K K[x]/(x^2)$. This is for example true for $Q$ being of type $A_1, A_2$ or $A_3$.

Question 2: For which combinations of $Q$ and $n$ is $KQ \otimes_K K[x]/(x^n)$ representation-finite?

I think question two has an easy answer for $n \geq 3$, namely only when $Q$ is of type $A_1$ for arbitrary $n$ or $A_2$ for $n=3$, but is there an elementary argument?