Hey, sorry for the late reply, I've been a bit pre-occupied today and I'm currently fairly drunk.
It's most tractable to use the Boyer-Lindquist form of the metric, for which ##g_{rr}## is seen to diverge whenever ##1-2m/r + a^2/r^2 = 0##. The roots ##r_{\pm}## of this equation correspond to the inner and outer event horizons. One can show that the 3-metric ##\gamma_{ab}## induced on the 3-surfaces ##\Sigma_{\pm}## (defined by ##r = r_{\pm}## respectively, i.e. just put ##dr=0## and ##r=r_{\pm}## in the 4-metric) has vanishing determinant, thereby implying that for any ##p \in \Sigma_{\pm}## there exists a non-zero vector ##v^a## such that ##\gamma_{ab} v^b = 0 \implies \gamma_{ab} v^a v^b = 0 \implies ||v|| = 0##, i.e. a tangent vector to null curves (photon orbits) lying entirely within ##\Sigma_{\pm}##; it should then be clear that null trajectories starting at less than ##r_{+}## do not ever breach ##\Sigma_+##, for example.
As for the ergospheres, they are defined as the region within which it's not possible to remain at fixed spatial ##(r,\theta,\phi)##, for which a necessary condition is that the "still" trajectory (with tangent vector ##u = \partial/\partial t##) is spacelike (and therefore unattainable), ##g_{ab} u^a u^b = g_{tt} u^t u^t > 0 \implies g_{tt} > 0##, i.e.\begin{align*}
\frac{2mr}{r^2 + a^2 \cos^2{\theta}} - 1 > 0
\end{align*}the roots (##r_{\pm}^{\mathrm{e}}##, say) of which define the inner and outer surfaces of the ergosphere.
