The odds that there is asbestos in the plaster are slim to none. Out of all the projects where I have worked on where there was asbestos abatement I have almost never seen it related to plaster. It is typically in pipe insulation, certain types of exterior siding, caulking and very commonly in floor adhesives used for VCT tile. Now on the other hand there is very likely lead from the paint used over the years and that is highly toxic as well when airborne.
The fact that the ceilings and moulding are falling down means that there is some serious issues with the ceilings. Is the wood lath falling off as well or just the plaster? I assume it's probably only the plaster as after time the keys that hold it to the lath break off and the plaster gives way. You should not be living under those conditions and the ceilings need to be fixed asap and the debris/dust cleaned up.
The easiest scenario for your landlord would be to install 1" X 3" wood furring on the celings screwed securely into the joists/rafters and hang drywall on the furring and finish.You will lose 1 1/4" of ceiling height and have to reinstall/replace the moulding but that's the best option. Gutting the ceilings is going to be very costly and should be avoided. Trying to salvage and fix the existing ceilings would be a waste of time and money IMO.
I would go under the assumption that there is lead present. Test kits at Home Depot are cheap and can confirm that. Your landlord needs to hire someone certified to deal with the lead/plaster. Most reputable contractors that work on homes built before 1977 are certified as that is a federal law and there are huge fines for those not certified. Your landlord should know lead is a huge liability in a home that age by now anyways and if they don't they better learn quick before they get sued.
Asbestos can definitely be found in plaster although his house may be a tad old for it.
The asbestos is put into plaster while it's wet?
I've honestly never seen it related to plaster either. IME, it's only as Rocket describes- used as fire-protection, insulation and also as acoustical underlayment (under wood floors in NYC c. 1920). I'm not the end all nose-sniffing pro like MC, just have to deal with it every project here in NYC- where I see it, it was installed as a blanket/wrap either around pipes or under floors. I haven't seen it or heard of it in plaster, or floating in the ceiling.
But I'd still get some tests done.
More importantly- looking to the contractor-guys out there- what caused the plaster to fall? I'm not used to stuff just failing without some source... and considering the falling has occurred at the same time, I'd think there was a leak, or shift in the building that caused the plaster to disengage from the wood lathe. Or maybe this happens naturally after a century plus? I dunno.