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.
Here's my take on drywall vs skim vs full update.
Skimcoat
Skim coating over the plaster that is already cracking would not be a good idea. Even if you use a harder more durable mud or plaster, it will still most likely crack because of the age of your house, and the fact that your walls are already cracking (and will probably continue to do so). The other down side is the sanding mess.
I recently bought a condo that had nasty texture as well on the walls which I wanted to get rid of. I decided for the first room that I would use a heavy 60 grit paper to remove as much of the larger chunky texture as I could. I then skim coated with drywall mud to smooth out the walls. It took several coats to get perfectly smooth, and a whole lot of cleaning afterward. But, in my case, I know my drywall is sound and not cracking underneath, so I could reliable coat without having to worry about patching too much in the future.
Drywall
Using a thin drywall layer over the top may be a quicker and cheaper solution. This will eliminate the cracking issues and will give you as smooth a wall as you like, with little mess in comparison to skim coating the entire wall.
I would still use a 60 grit paper or paint scraper (if it will cut through your texture) and knock down as much of the heavy texture to allow the drywall to sit closer to the surface. This will help eliminate waviness in the wall where the texture might be outrageous (like it was in my house), and allow you to use a thinner drywall.
A possible major downside to this is electrical boxes? Maybe someone with more electrical knowledge can speak whether adding 1/2 inch thickness to a wall would cause issues with the electrical boxes being set further in? If so, again, I would sand and use 1/4 inch.
Full Update
If I was going to be living in this house for the rest of my life, I would consider a full update, especially if I was already going to be making changes to wiring.
This solution is costly, time consuming, and messsssy! But, the upside is that you can fix/update your electrical to be safer, and more suitable to modern appliances. You can also add low voltage through the house which (if you're a tech nerd like me) may be a nice plus. This will also give you perfectly smooth walls that won't crack and a feeling that your not hiding a mess in your walls.
Hope this is helpful!
Best Answer
How do you plan on not painting if the new trim is lower than the old trim? Personally, I'd go with the same if not larger trim to hide any imperfections left by the old trim, but maybe that's just me.
To remove old caulk, just use the flat edge of the painters multi-tool to scrape it away. Be careful not to dig into the wall or pull up the paper from the drywall. For anything left, sand it down. You'll probably want to have some extra drywall mud or spackle available to fix any spots and to cover any screw/nails that were previously covered by the old trim.
And then, yes, you'll need to paint. It's easier to do a first coat on the trim and the walls before installing the trim. Then all you have to do is touch-up any nail holes or scrapes.