If the laths are damp and your plasterboarding, remove all the old laths first, messy job but will make the boarding easier to do.

1) Yes do not plasterboard over damp wood/laths as movement may often occur causing cracks and possibly instability in the fixings.
2) Remove all the plaster not only to expose the extent of which you should expose about 300mm more than the problem. But you will need to remove any plaster that contains hygroscopic salts which will continue to attract moisture from the air after the water ingress problem is fixed.
Note: Water ingress doesn't describe the actual problem? It can occur from leaking drainage, plumbing etc. or be water under pressure or gravity.
Sometimes fixing the latter can move the problem elsewhere!
Correct diagnosis, preparation, physical barrier and salt prevention in the finishing process is necessary.
Oh, and a 20yr guarantee!
Hope this helps.

remove all plaster.normally lathes only take 48hrs to 72hrs to dry out.if time is a factor, tack a vapour barrier up or use mositure board.

all damp laths will need to be removed as will all old plaster before u plasterboard the ceiling also keep in mind that if the plaster is wet and the larths are wet then the joists may also be wet or damp and will need to be allowed to dry out before boarding removing an old lime ceiling is a very dirty job and very dusty you will need a dust mask and not one of those cheap paper ones hope this helps and good luck