Ceiling – Can we plaster our ceiling without removing the crown molding

You can drywall over the plaster, but you'll have to do the entire wall and screw through the plaster into the lath/studs.

Alternatively, you can fix the plaster itself. The problem is, it's separated from the lath. The fix for this is to drill shallow holes into the affected area JUST DEEP ENOUGH to penetrate the plaster, not the lath. Drill the holes 4" apart in a grid pattern.

Then use a caulking gun and inject construction adhesive into each hole to create a glue layer behind the plaster between it and the lath.

Press the plaster up against the lath with a flat board, holding it ONLY UNTIL THE PLASTER STICKS. Don't leave the board there too long or the glue will grab it too. Slide the board up and down to keep it from sticking while the glue takes hold between the lath and plaster.

Allow it time to dry, then patch the holes and paint.

The fact that sections are peeling off indicates that there is a structural problem with the overall "sandwich."

In any repair, unless you can overcome the structural deficiencies with an overriding mechanism, you have to strip back the unsound sections until you get to good structure. With you description , it is hard to see where that is.

You could knock loose all the compromised sections and then try to patch the open areas, but it sounds like the overall lathe-plaster-chicken wire-plaster amalgam may be questionable. You may do this and shortly thereafter find a new section that is failing.

I think that a drywall overlay may be the simplist, fastest and cheapest solution, by far. Whether you strip down to the chicken wire or to the lathe depends on how much of the ceiling is loose/compromised. If height is not a problem, dislodge whatever is loose, use shim to insure level and then screw up thin drywall over the whole ceiling. DO NOT USE NAILS! If you hit studs, great, but the lathe should be adequate to hold a thin wallboard lamination.