As long as you can fix the plasterboard ok I don't see why not. I've just put some up in the toilet room where the roofer put his foot through - used plasterboard adhesive and some screws to hold it firm.
Yes. Just use the longer screws and make sure your screwing into joists and noggins and not just into the laths.
Ideally you want to remove it all and in places you will need to anyway (unless you have access to the joists from above the ceiling. I very doubt there will noggins and joists in the right places where you require plasterboard fixings so you will have to remove some to fit timbers to support the boards all around.
I've just spoken to my plasterer about this as I have a job which needs a new ceiling - he said it isn't worth the aggro of removing the old lath & plaster ceiling, and he [nearly] always just overboards.
Both methods 1. Using plasterboard over Lathe 2. Strip back to uprights. Both depend on the job you want and if you have money or desire to do a refurb. One is cheaper first option and the second to solve other issues.
Lathe removal is dirty/dusty/from ceiling you will probably have ashes as deafening and "tarpaulin is us" all the room up to proceed or if your lucky you could be that Holmes guy on Sky with an army of lasses, wonder what his job cost and if he would get work if it was not just for a TV programme with all them builders he has in tow. Never seen so many contractors on a job for the size. Back to removal the Lathe upright centres depended on lathe maker and length of stock materials are all over the place. So do not expect 400mm centres, do expect half laps joints/wet rot/dry rot/woodworm/damp/materials not straight or even wet rot dried out so to speak, even had fleas infestation in the horse hair had to bomb the place twice. That is why its best to remove depending on what affects your under lying issues you may be affected by, cheaper in those scenarios to do a strip out. As a straight fix for some clot putting his foot through first option above is good choice hope he made good at no cost to you as well. Your lucky no elaborate plasterwork got trashed, seen whole ceiling falling down, that was a mess and the price to replace was a bit eye watering. Had to make some jigs for a left handed (they were always handed for the eventual user in my experience) plasterer for shooting boards think that is what they were called.
Glad it all worked out or you could have been Charlie Brown mate with the cloud following him around if you had to remove the lathe, its real surprising how far that fine dust travels all over the house but Holmes has the answer "lasses with hoovers" in make up as well and rather low cut tops, my lads would never get no work done with that full buxom look and stooping/bending over. It would end in a divorce Canadian Male Builders must be made from granite or is Holmes guys specially selected.