Let $X$ be the class of (unital, associative and not necessarily commutative) rings $R$ where every indecomposable projective $R$-module is finitely generated.

Question 1: Is there a nice equivalent characterisation when a ring is in $X$?

Question 2: $X$ should contain for example all Artin algebras. Does it also contain right artinian rings and if yes, is there an easy argument?

When every projective module is a direct sum of finitely generated modulesby McGovern, Puninski and Rothmaler. But it seems quite complicated, and there are probably even more exotic examples in your class $X$: e.g., I don't see why there shouldn't be rings where not every projective is a direct sum of f.g. projectives, but there are no indecomposable projectives at all. $\endgroup$