Question number 1: Let your base S be Spec k[x] (say k is an algebraically closed field), let X be Spec k[x,y], and let G be Ga,S, with action over the point s given by gs(xs) = (sgs) + xs. This action is transitive away from zero, so the orbit of the zero section is a plane with a slit. This is not an open subscheme of a closed subscheme of X, because it is not a scheme.
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Let your base S be Spec k[x] (say k is an algebraically closed field), let X be Spec k[x,y], and let G be Ga,S, with action over the point s given by gs(xs) = (sgs) + xs. This action is transitive away from zero, so the orbit of the zero section is a plane with a slit. This is not an open subscheme of a closed subscheme of X, because it is not a scheme. |
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