Fixed loci are closed and a normal subgroup fixing a point fixes the orbit closure

For each $h\in H(k)$, its action and the identity are $k$-morphisms $X\to X$. Their scheme-theoretic equalizer is closed by [F2]. The schematic intersection of these closed equalizers has exactly the $k$-points fixed by $H(k)$, so these form a closed subset of $X(k)$. Topological invariance of a nonclosed point alone does not imply membership in a scheme-theoretic equalizer. Only $k$-points are used to define these $k$-automorphisms; an arbitrary $R$-point would act on $X_R$. This proves (a).

The closed subscheme $H\cap G_y$ contains every point of $H(k)$, since these fix $y$. Smoothness of $H$ and [F3] therefore give $H\subseteq G_y$ scheme-theoretically. Consequently every point of $H(R)$ fixes $y_R$ for every base algebra $R$. Normality then gives $h(g{y}_R)=g(g^{-1}hg)y_R=g{y}_R$, so $H$ fixes the entire orbit morphism on every base algebra. The closed locus in step1.1 contains the orbit and hence its closure $Z$.

By [F4], $Z$ is the reduced scheme-theoretic orbit closure. The composite $G\times G\to G\times Z\to X$ given by action equals the orbit morphism after multiplication and factors through $Z$. The map $G\times G\to G\times Z$ is schematically dominant by [F4], so the pullback of the closed ideal defining $Z$ vanishes already on $G\times Z$. Thus the action factors through $G\times Z\to Z$, proving scheme stability. Similarly, the two maps $H\times Z\to X$ given by action and projection agree after the schematically dominant $H\times G\to H\times Z$, by step2.1. Their closed equalizer [F2] must therefore be the whole $H\times Z$. Hence $H$ fixes $Z$ scheme-theoretically, in particular pointwise, proving (b). No reducedness of ambient $X$ is needed.

For any $x\in Z(k)$ fixed by $H(k)$, the closed subscheme $H\cap G_x$ contains $H(k)$. Smooth-point density [F3] makes it all of $H$, so $H\subseteq G_x$, proving (c). Steps1.1,3.1 and4.1 establish the three claims, with the all-base-algebra argument in step2.1 licensed by the preceding scheme-theoretic stabilizer inclusion.