As shown in the figure, in the quadrangular pyramid $P-ABCD$, $AB \parallel CD$, and $\angle BAP = \angle CDP = 90°$.
(1) Prove: plane $PAB \perp$ plane $PAD$;
(2) If $PA = PD = AB = DC$, $\angle APD = 90°$, and the volume of quadrangular pyramid $P-ABCD$ is $\frac{8}{3}$, find the lateral surface area of this quadrangular pyramid.