mg.metric geometry - Can any rectangle be inscribed in any convex figure?

Yes, this follows from a more general result in

Nielsen and Wright, Rectangles inscribed in symmetric continua. Geom. Dedicata 56 (1995), no. 3, 285–297 MR

(This is reference 4 in the Wikipedia article I quoted in my answer to your previous question.)

In their terminology, a simple closed curve $C$ is symmetric if there exists a point $Pnotin C$ such that each straight line through $P$ intersects $C$ in exactly 2 points. This condition is trivially satisfied when $C$ is a boundary of a convex region.