1. If $\theta$ is an involution of a complex linear algebraic group $G$ and if $K=G^\theta$ is its fixed-point set, then $K/K^0$ will always have exponent 2. This follows from a generalized "Cartan decomposition". The argument goes as follows. Let $\mathfrak p$ denote the (-1)-eigenspace of $d\theta$ on $\mathfrak g$. Then, morally at least, we ought to be able to express everything in $K$ as $k=k_0\exp X$, with $k_0 \in K^0$ and $X \in \mathfrak p$. But then $k^{-1} = (k_0\exp X)^{-1} = \exp(-X) k_0^{-1} = k' \exp(-X)$, for some $k' \in K^0$ (as $K^0$ is normal in $K$). On the other hand, $\exp(-X) = \exp(d\theta(X)) = \theta(\exp X) = \exp X$. Consequently, $k = k^{-1}$ in $K/K^0$, as desired. I've glossed over some details, which you can find in Loos, _Symmetric Spaces_, vol. 1, Benjamin (1969). 2. This follows from Theorem 8.1 in Steinberg, _Endomorphisms of linear algebraic groups_, Mem. Amer. Math. Soc. **80** (1968). An older reference---for compact Lie groups, at least---is Theorem 3.4 in Borel, _Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes_, Tôhoku Math. J. **13** (1961), 216–240.