Claims About a Perpendicular Pyramid

In a tetrahedron $P\text{-}ABC$, $PB=PC$, $AB=AC=\sqrt 2$, $AB\perp AC$, and the plane $PBC$ is perpendicular to the plane $ABC$. Let $V$ be the volume of $P\text{-}ABC$ and $r$ the radius of its inscribed sphere. Determine, with proof, which of the following claims are true.

1. The dihedral angle $B\text{-}PA\text{-}C$ is greater than $\dfrac{\pi}2$. 2. The dihedral angle $A\text{-}PB\text{-}C$ is less than $\dfrac{\pi}4$. 3. $r<\sqrt 2-1$. 4. $\dfrac 2r-\dfrac 1V\geqslant\sqrt 6+2$.