A Pyramid in a Cube

A regular quadrilateral pyramid $P\text{-}ABCD$ has base edge $1$ and height $h$, and its apex $P$ lies inside (or on the surface of) the cube $ABCD\text{-}A_1B_1C_1D_1$. Determine, with proof, which of the following are true:

1. the range of $h$ is $(0, 1]$; 2. if the lateral edge of the pyramid has length $\dfrac{\sqrt{3}}{2}$, then $h = \dfrac{\sqrt{2}}{2}$; 3. when $P$ is the center of the top face $A_1B_1C_1D_1$, the circumscribed sphere of the pyramid has surface area $\dfrac{9\pi}{4}$; 4. when $P$ is the center of the cube's inscribed sphere, the common part of that inscribed sphere and the pyramid has volume $\dfrac{\pi}{36}$.