Let $\frak g$ be a semi-simple finite dimensional Lie algebra over the complex numbers $\mathbb C$. Then every non irreducible representation of $\frak g$ is completely reducible.
Q1: Is category f.g. left $U(\mathfrak{g})$-Mod is equivalent to the category finite dimensional $\operatorname{Rep}(\mathfrak{g})$?
Q2: If every f.g $U(\mathfrak{g})$ module is completely reducible (in other words, they can be written as direct sums of simples) and above category equivalence holds, then from completely reducibility, I deduce for semi-simple Lie algebra, $U(\mathfrak{g})$ is semisimple as a ring. Hence, I would expect all $U(\mathfrak{g})$ modules to be projective and injective at the same time. Is this correct?
Q3: If Q1, Q2 are wrong, what is the relationship between semi-simple Lie algebras and semi-simple rings? (I would not expect for infinite dimensional representation, above categorical equivalence holds. The "semi-simple" ring should be interpreted as every f.g. module is decomposed into simple modules only.)