Adding 2X4 ceiling under lath and plaster

Removing plaster can make tremendous amounts of dust, isolate the area with plastic to cut down on the spread, and get yourself a dust mask and a helmet is a good idea. Cover any cold air returns and heating vents. Once you're prepared, taking plaster walls down is pretty easy, whack it with a hammer to bust a hole, then you can usually get a shovel or pitchfork behind the plaster and it will pull away from the lathe easily as you break the keyway that holds it in. Clean up all the plaster and dust, then go after the lathe with a crowbar and/or hammers; then start cutting the vertical studs with a sawzall.

Double, triple, quadruple check that it's not load-bearing, if the blade is binding when cutting through a stud, there's downward pressure, and even if it's not structurally load bearing, things could've shifted over time above, and taking it out might make things drop or sag.

Most likely it is shiplap. We just remodeled our kitchen in our 1935 house and underneath the drywall and paperboard, we had something similar. Actually the order was: shiplap, cheesecloth, several layers of wallpaper, then either paperboard (lower half) or drywall (upper half).

As far as whether or not it is structural, we went with the assumption that it was. I'm sure the vertical spacing of the studs in approach does not meet current building standards. However, I am not an architect either. At the very least, having it behind the drywall makes it incredibly easy to hang the drywall and anything else for that matter.