I would like to take any shape, and drag its edges wherever I want, distorting the image in the process. How do I achieve this ? Perspective isn't what I am looking for, nor is Filters>Distortions>Curves.
This looks like the Cage tool, once you have put nodes around the shape.
Perhaps you are looking for the Unified Transform Tool Shift+T.
It's a multi-functional transform tool. Hold down Shift as you click and drag an edge handle to distort the image, squishing or stretching it. Moving the edge without Shift resales proportionally. The corner handles can distort like perspective, and the white diamond handles are for skewing.