In the paper

Arkowitz, Martin Categories equivalent to the category of rational H-spaces, Manuscripta Math. 64 (1989), no. 4, 419–429

it is shown that the rational homotopy equivalence $G_\mathbb{Q}\cong\prod_n K(\pi_n(G_\mathbb{Q}),n)$ is an equivalence in the category of rational $H$-spaces if and only if $G$ is homotopy abelian (see Proposition 3.1). This means, I think, that the answer to your first question is no in general.

However I think the answer to your second question should be yes, but for different reasons. If $G$ is a connected topological group of the homotopy type of a finite CW complex, then theorems of Hopf and Borel imply that $$H^\ast(BG;\mathbb{Q})\cong\mathbb{Q}[y_1,\ldots , y_k]$$ where each generator $y_i$ is of even degree (see McLeary's A User's Guide to Spectral Sequences, Theorem 6.38). It follows that the minimal model of $BG$ is evenly generated, and has zero differential, meaning that $BG$ is rationally a product of Eilenberg-Mac Lane spaces.

In the paper

Arkowitz, Martin Categories equivalent to the category of rational H-spaces, Manuscripta Math. 64 (1989), no. 4, 419–429

it is shown that the rational homotopy equivalence $G_\mathbb{Q}\cong\prod_n K(\pi_n(G_\mathbb{Q}),n)$ is an equivalence in the category of rational $H$-spaces if and only if $G$ is homotopy abelian (see Proposition 3.1). This means, I think, that the answer to your first question is no in general.

However I think the answer to your second question should be yes, but for different reasons. If $G$ is a connected topological group of the homotopy type of a finite CW complex, then theorems of Hopf and Borel imply that $$H^\ast(BG;\mathbb{Q})\cong\mathbb{Q}[y_1,\ldots , y_k]$$ where each generator $y_i$ is of even degree (see McCleary's A User's Guide to Spectral Sequences, Theorem 6.38). It follows that the minimal model of $BG$ is evenly generated, and has zero differential, meaning that $BG$ is rationally a product of Eilenberg-Mac Lane spaces.