Three inch gaps seem quite large - even for a 300 year old wall.
You just need to use a good mortar mix (sand and cement) on the narrower gaps.
Clean out all the old, lose material first to make sure that you get good adhesion and then push the mortar right into the gaps as far as it will go. Work on a small area (1 m2) at a time.
Adding pea gravel to give the mix more "body" might work on the wider ones.
However, I'd get a professional to have a look at the wall first to make sure you're not going to have to redo this work in a couple of years time. If the foundations have gone it might need something more substantial work doing.
Is the wall part of a house or just a garden wall? If it's the latter then dismantling and rebuilding might be the best long term option.
Sounds like dehumidification might be the answer. The cause of the dank "basement" smell, is high humidity. Removing moisture from the air is a side effect of refrigeration, which is why the air conditioner helps. A dehumidifier is just like an air conditioner, except that the dehumidifier heats the air back up after cooling it and removing moisture.
You may be able to simply use a standalone dehumidifier, though then you'll have to dump the collection bucket. Another option, is to install a whole house (or whole apartment) dehumidifier. A whole house unit will be more expensive, but won't require dumping a bucket or water once a day.
If you have moisture problems in the basement, you'll want to sort those out. That should eliminate the cause of the foul air, and the need for removing it.
If you don't want to deal with the moisture problems, then you'll have to focus on removing the foul air and replacing it. This means drawing in outside air, conditioning it (heating, cooling, filter, humidify, dehumidify, etc.), and exhausting the foul air.
If you live in a mild climate where indoor and outdoor temperatures are similar, this shouldn't be that bad. However, if you live where indoor and outdoor temperatures are extremely different, you're going to be paying a lot to condition the air.
You'll want to look in to a heat recovery ventilation (HRV), and/or energy recovery ventilation system. As these systems will help reduce the amount of heating/cooling that will have to be added to the incoming air.
Drawing air from one dwelling unit into another, is not a proper solution. Air must be moved within a single dwelling unit, or between the inside and out.
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I think you may need a whole bunch of new ones, or a completely different ceiling system (as you prefer). Those should be attached to the ceiling structure, and the backside of the fallen one looks like parts that should support it are ripped away.
It's been a long time since I have used any of these - I vaguely recall that they would latch into others on two sides, covering the fasteners (staples, usually) of the ones they locked into, get stapled on the exposed sides, and have the next tile cover up their staples. So your expectation of having to work to a wall sounds about right.
A grid-type suspended ceiling system is similar, but gives better access above it if needed as the panels just sit in the frame; though it takes up a bit more headroom (which you might not want in a basement.) You could drywall and have similar height (it may be "louder", though), or you can find this type of acoustic tiles (but you're probably going to have to replace them all, for color & pattern match if nothing else. 30-40 years of aging shows when you put a new one right next to it.)