Drywall – required to fix this plastered drywall ceiling

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.

What is the thickness of that second layer of drywall in your great room? Chances are it's very thin, which you could hide by simply fanning out your joint compound to make a "ramp" 1 1/2 to 2 feet wide where the kitchen meets the great room. Nobody would ever be able to see the difference. Another option is to put a fake beam between the two rooms - basically a thin piece of wood trim that looks like a fancy support beam partially buried in the ceiling. Again, nobody would ever notice the difference in the ceiling heights between the two rooms.

There is a VERY easy way to get rid of the popcorn ceiling. Get a paint roller with a fairly thick roller sleeve and put it on a long handle. Dip the roller into a bucket of water, and then "paint" the popcorn ceiling with the water. Use a bit of pressure to actually get the water into the popcorn, but don't overdo it and soak your drywall underneath. Let that soak in for a minute or two, and then the popcorn will come right off. Every time I paint a popcorn ceiling, half the popcorn wants to fall off with the roller, so I tried simply rolling it with water when I wanted to remove the popcorn one time, and it worked great.