Continuing the investigations started in the recent work [12] on semi-global controllability and stabilization of the (1+1)-dimensional wave maps equation with spatial domain $S1$ and target $Sk$ , where semi-global refers to the $2π$-energy bound, we prove global exact controllability of the same system for $k > 1$ and show that the $2π$-energy bound is a strict threshold for uniform asymptotic stabilization via continuous time-varying feedback laws indicating that the damping stabilization in [12] is sharp. Lastly, the global exact controllability for $S1$-target within minimum time is discussed.