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.
Yeah it is more brittle than sheetrock but still strong stuff. Start with scoring the paint/caulk around the casing with a utility knife, then use a pry bar as normal, except use a board or plywood between your prybar and the wall. That way you won't have the pry bar indent into the plaster as you pull the wood off. Work your way around the door casing, prying it a little at a time. Once you see where the nails are you can pry directly under that point, or hit the face of it with a hammer and sometimes that makes the nails pop out enough you can pry them out through the face.
Best Answer
You cannot identify asbestos-containing material by looking at it.
If in doubt have it tested!