Suppose $G$ is an algebraic group with an action $G\times X\to X$ on a scheme. Does the fixed locus (the set of points x∈X fixed by all of $G$) have a scheme structure? You can obviously define the functor $\operatorname{Fix}(T)=\{t\in X(T)\mid \text{$t$ is fixed by every element of $G(T)$}\}$. I...

The question gives the "wrong" definition of $\operatorname{Fix}(T)$, hence the resulting confusion. A more natural definition of the subfunctor $X^G$ of "$G$-fixed points in $X$" is $$ X^G(T) = \{x \in X(T) \mid\text{$G_T$-action on $X_T$ fixes $x$}\} = \{x \in X(T) \mid\text{$G(T')$-action on...