Drywall – 1/4“ to 1” gaps in corners of the newly “professionally” installed drywall

Since you want to have the drywall piece be 5" high I think you will want to have backer behind it for its full height. Consider making it 5.5" high and then using a piece of 2x6 lumber (that comes 5.5 inches wide) as the backer piece. Cut a couple of pieces to straddle across the bottom of this opening at the back and middle to hold up the shelf the full length. To install the front backer piece pre-drill clearance holes through the 2x6 that let your screws slide in place. Then use screws about 3" long to secure through the existing drywall at the corners. This will end up surprisingly strong! See picture below.

The trick to get the 2x6 to screw in place would be to start two screws from each end on each side (eight screws total) and tighten them down evenly and gradually. Make sure to stagger the pre-drilled holes from side to side of the 2x6 on each end so that the screws do not collide with each other as they enter the existing corner studs.

One of the main purposes of moldings, such as the trim around doors and windows (called casings), is to act as a barrier and seal to wind and water intrusion. Moldings on wall between vertical boards, called battens served a similar purpose. The decorative element was an extra benefit (unless you are an extreme modernist/minimalist who wants totally flat featureless transitions).

Since the days of rough hewn houses, our water and wind protection has improved overall. However casing still are a significant protector, especially against air leaks in cold climates.

Casings provide the protection in two ways - they create a convoluted path for any air to get through by overlapping both the door frame (jamb) and the wall material (plaster, wallboard, paneling, etc); and they they fit tightly and are generally sealed to the other two parts with paint and often with caulk.

Your first approach will compromise both of those features. Your are effectively creating a butt joint with no overlap. The path for air is directly perpendicular to the wall and the casing. The joint will likely crack at some point, even if you mud or caulk. Those seals work well when they are used in a corner that meets at a right angle, with one member going behind the other, not a butt joint. If you add an additional molding, you still have a butt joint along the edge of the original casing that might crack, and you are complicating your profile.

The second approach solves that problem, but getting the old plaster and lath out, and getting a tight fit pushing forward against the in-place casing will be hard.

The third approach seems much harder, but actually may be easier and less time consuming than the first or the second. Once the casings are off, removing the plaster remnants will be much easier. If you carefully score the joints around the casings and where they meet at the top with a utility knife, pry them off using a small catspaw type prybar, pull the nails through the back of the casing, lightly sand the chipped paint off the edges before reattaching, it should be quite manageable. It sounds harder than it is.

Even if you were to crack a molding or two in the process, they can be glued back together, and once painted, will seem whole.

P.S. My house was built in 1869, renovated in 1905, and renovated again in 1999. In the last reno, the above process is exactly what was done, leaving great early 20th century details, smooth walls and tight seams.