Let $G$ be a group scheme over $S$ where $S$ is a reduced scheme of finite type over a field $k$ of characteristic 0, and let every fibre $G_s$ over a closed point of $S$ be isomorphic to $\mathbb{G}_a^n$ for some $n$ that varies with $s$.

~~Is it true then that $G$ is reduced?~~

Suppose that $G$ is reduced. Is it true that $G \times_S G$ is reduced?

(This question is a continuation of this one; the motivation comes from this question)