Quadratic Stark Effect - Perturbation Theory

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Homework Statement

Homework Equations

The Attempt at a Solution

With a parity operator, Px = -x implies x has odd parity while Px = x implies x has even parity.

Things that puzzle me

1. Why is $[H_0,P] = 0$ and $H_1P = -PH_1$? Is it because $H_1 \propto z$ so $Pz = -z$? Then shouldn't it be $PH_1 = -H_1$?

2. For any operator R, it is represented by a matrix $R_{ij} = <i|R|j>$. In this case is the operator in question $PH_1 + H_1P$? What does 'matrix element between two parity states' mean? From what I see, $<n'l'm'p'|PH_1 + H_1P|nlmp>$ is simply the addition of two matrices, one corresponding to $PH_1$ and another $H_1P$. Which has odd/even parity and why?

3. Why do all diagonal matrix elements vanish?

 

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Would appreciate clarifying doubts on the 3 points above!