Let $G$ be a finite group of Lie type. By Deriziotis' and Carter's articles we know that conjugacy classes of connected centralizers of semisimple elements are parametrized by $(J,[w])$ where $J$ is a subset of a certain set of roots of $G$, with respect to a maximal torus contained in a Borel subgroup, and $[w]$ is a conjugate class in $N_{W}(W_J)/W_J$, where $W$ is the Weyl group of $G$, and $W_{J}$ is generated by reflections corresponding to $J$. My questions are the following:

If $C_{G}(s)$ is connected and corresponds to $(J,[w])$, then $C_{G}(s)$ is contained a standard parabolic $P_{J}$. Is it true that if $[w]\neq1 $, then $C_{G}(s)$ is contained in a Levi subgroup of $P_J$?

If $C_{G}(s)$ and $C_{G}(s')$ are corresponding to $(J,[w])$ and $(J,[w'])$ respectively with $[w]\neq [w']$, can we say that their root systems and their Weyl groups are equal?.

Assume that $C_{G}(s)\subset C_{G}(s')$, with corresponding conjugacy classes $(J,[w])$ and $(J',[w'])$. Can we conclude that $J\subset J'$?