How to repair this ceiling mounted cloth drying rack?

On numerous occasions I have had to cut hollow core doors off more than the available blocking in the top / bottom of the door. What I have done in these instances is to take the cut off bottom piece and then reclaimed the inner block for re-use. If a table saw is available it is quick work to set the fence just right so that a couple of rip direction passes cleanly cuts off the outer laminations. Alternatively the laminations can be removed using a plane.

Once the block is nicely cleaned up it can be glued and slipped right up into the open bottom of the door. Use a couple of 1x4 boards on either side of the door and some good sized C-Clamps to tighten up the glue joint so that the glue can dry without any gaps.

Note that often the inside of hollow core doors has an internal webbing of cardboard material to bridge the gap between the two lamination skins of the door. Sometimes it is necessary to use a chisel inside the cut open bottom of the door to scrape back the glue joint of this web material on the laminated panel. If this is not smoothed out the re-claimed bottom block will make the laminated sides bulge out when trying to glue the block in place.

Go ahead and place the drywall over the top edge of the crown moulding. Sounds like the wall was already re-plastered with it in place. If you tried to remove it, you may end up re-doing the walls as well. Even forgetting that, it's almost impossible to remove moulding without sustaining some loss. Without replacement material available, removal with intents of preservation is extremely risky, especially with stain grade material. You don't want to go there.

If your joists were flat, you'd still only need to shim down the first 4 feet or so along the walls, no one will see the resulting slight slope. Keep this in mind when planning your shims. It doesn't have to be perfectly flat, unless that gives you pleasure. You also don't need to shim to completely fill the gap. An 1/8" gap makes a clean shadow line that is not detrimental (assuming it is perfectly uniform) at all. It also makes finishing easier. The taped joints of drywall are normally not perfectly flat, despite the tapered edges. There is a slight bulge which is usually invisible. But against straight moulding, depending on the profile, this bulge may be visible. You might consider grinding down the edge tapers at the board corners against the wall so the joint tape can be completely buried and the yet the joint against the moulding can still be perfectly flat.