Let $k$ be a field. Consider a group $k$-scheme $G$ and let $X$ be a $k$-scheme equipped with an action of $G$. Then one can define the quotient stack $[X/G]$. Objects of $[X/G]$ over $k$-scheme $T$ are pairs $(\pi, \alpha)$ such that $\pi:P\rightarrow T$ is a locally (with respect to fpqc topology) trivial $G$-bundle and $\alpha:P\rightarrow X$ is a $G$-equivariant morphism.

Now in Olson's book in Example 8.1.12 the author assumes that $G$ is smooth in order to derive that the canonical map $X\rightarrow [X/G]$ is smooth and in result to infer that $[X/G]$ is an algebraic stack. It seems that the other part of his argument, which shows that the diagonal $\Delta_{[X/G]}:[X/G]\rightarrow [X/G]\times_k[X/G]$ is representable holds for any group $k$-scheme.

Here are some immediate questions.

1. Under what conditions on $G$ the stack $[X/G]$ is algebraic? Is smoothness essential?
2. If $G$ is affine over $k$, then is $\Delta_{[X/G]}$ representable by quasi-affine morphism of algebraic spaces? If not, then what one should impose on $X$ to know that this is the case?