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
In their terminology, a simple closed curve $C$ is symmetric if there exists a point $P\notin C$ 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.