Properties of a Reuleaux Tetrahedron

A Reuleaux tetrahedron is the solid obtained by intersecting the four balls whose centers are the vertices of a regular tetrahedron and whose common radius equals the tetrahedron's edge length. Consider the Reuleaux tetrahedron built on a regular tetrahedron $ABCD$ with edge length $4$, as shown.

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

1. If $P$ and $Q$ are any two points on the surface of the Reuleaux tetrahedron, then the maximum value of $PQ$ is $4$. 2. The cross-section of the Reuleaux tetrahedron cut by the plane $ABC$ has area $8(\pi-\sqrt 3)$. 3. The volume of the Reuleaux tetrahedron is $8\sqrt 6\,\pi$. 4. The radius of the inscribed sphere of the Reuleaux tetrahedron is $4-\sqrt 6$.