Folding a Right Triangle Along the Median to the Hypotenuse

In right triangle $ABC$, $\angle B=\dfrac{\pi}{2}$ and $AC=2BC=4$, and $D$ is the midpoint of segment $AC$. Triangle $ABD$ is folded along $BD$, carrying $A$ to a point $P$, so that a tetrahedron $P\text{-}BCD$ is formed (left: before the fold; right: after the fold).

Determine, with proof, which of the following claims about the folding process are true.

1. The circumradius of triangle $PBD$ is $2$. 2. In some position of the fold, $PD\perp BD$. 3. In some position of the fold, $PB\perp CD$. 4. If $PD\perp DC$, then the circumscribed sphere of tetrahedron $P\text{-}BCD$ has volume $\dfrac{32}{3}\pi$.