ag.algebraic geometry - Is every homogeneous G-variety of the form G/H?

It depends on what you mean by "closed subgroup". If you mean a Zariski closed subset which forms a subgroup then the answer is no. If you mean a closed subgroup scheme, then the answer is yes. An example where you need to use the second definition is the Frobenius map $Fcolon G to G^{(p)}$. If we let $G$ act on $G^{(p)}$ through $F$ then the action is transitive and indeed $G^{(p)}$ is isomorphic to $G/Ker F$. However, unless $G$ is zero-dimensional $Ker F$ is a non-trivial finite group scheme whose $k$-points consist of just the identity.

Note however that $G/H$ is always quasi-projective even when $H$ is a subgroup scheme so all homogeneous $G$-spaces are quasi-projective.

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