For (2): let $G=\mathbb Z$, so $\phi$ is generated by a single homeomorphism of $X$.  Let $X=\{ z\in\mathbb C : |z|\leq 1\}$ and define $\phi(re^{i\theta}) = r^2 e^{i\theta}$.  Then $0$ and all points on the circle are fixed; the orbit of any other point $re^{i\theta}$ has accumulation points $0$ and $e^{i\theta}$.  Let $f\in C(X)$ be invariant; translate so $f(0)=0$.  Then $f(re^{i\theta}) = \lim_n f(r^{2n}e^{i\theta})=0$ for all $r<1$, so by continuity $f=0$.  Thus the action $\alpha$ is "ergodic", but $\phi$ has non-trivial invariant open and closed sets.