For 2), consider the circumscribed circle of the $ABC$ triangle, and expand it to a sphere in 3d with the same center and radius. If we cut it by a plane orthogonal to the $ABC$ plane and containing $AB$, we get a circle with $AB$ as its diameter, so, by Thales' thm the points $M$ and $P$ will be on that circle, hence on the sphere. Similarly for $N,Q$. Well, it's still needed that these 4 points are in the same plane..
So, for the rest, I would use vectors (or coordinate geometry), setting up the coordinate system in a preferable way, say $A$ is the origo, $ABC$ plane is the $x,y$-plane, we can set also $AB=(1,0,0)$ if it helps..