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edited title

Relation between finite dimensional representations of an affine group scheme and quasicoherent sheaves on the classifying stack

Let $G$ be an affine group scheme over a field $k$ of characteristic zero.

I understand that when $G$ is algebraic there is an equivalence between the category $\text{Rep}(G)$ of (rational) representations of $G$ and the category $\text{QCoh}(BG)$ of quasicoherent sheaves on the fppq classifying stack $BG$.

I have three questions:

  1. Which subcategory, if any, of $\text{QCoh}(BG)$ corresponds to the category $\text{Rep}_{f}(G)$ of finite dimensional representations of $G$? Does it makes sense to talk about coherent sheaves on $BG$?
  2. Does an equivalence of this form hold when $G$ is not assumed to be algebraic?
  3. How about at the level of bounded derived categories?