AC
From $\angle APB=120°$ and $AP=2$, we know the base diameter $AB=2\sqrt{3}$ and height $PO=1$. Thus the volume of the cone is $\pi$, so A is correct. The lateral surface area of the cone is $2\sqrt{3}\pi$, so B is incorrect. Connect $CB$ and let $Q$ be the midpoint of $AC$. Connect $QO$ and $PQ$. It can be easily shown that the planar angle of the dihedral angle $P$-$AC$-$O$ is $\angle PQO=45°$, so $QO=PO=1$ and $PQ=\sqrt{2}$. Thus $BC=2$, so $AC=2\sqrt{2}$, and C is correct. $S_{\triangle PAC}=\frac{1}{2}AC\cdot PQ=2$, so D is incorrect.