First off, I believe this documentary will prove beneficial for you: https://www.youtube.com/watch?v=yVkyfC_AOLI
To answer all your questions, given near unlimited resources you can build a construction to house an arbitrary number of people underground. However, if your resources are limited, it depends on a bunch of things, namely the following:
1- How do you intend to configure the underground areas to house everyone. What type of geometry, living spaces and what not are you picturing. This will be a major influence on whether it's feasible or not.
2- At what rate do you want to expand the habitation at.
3- How far underground you want your colony to be.
4- The geology which you are mining through.
In response to your initial questions:
It occurred to me, perhaps incorrectly, that a softer stone like limestone might allow them to do this more easily
This ones a double edge thing, mining in soft rock is much easier than in hard rock, although the major downside is that soft rock isn't as strong as hard rock. The trade off would be: harder to dig but much bigger/safer rooms or easier to dig but necessarily smaller/slightly less safe rooms.
At what point, either depth or size, would it become too unstable to risk living in? Or is that not how stone works?
That's not how stone works. Given the proper data and knowledge, a suitable habitation can be engineered in any rock mass, however the cost of doing so (resources, time, etc) will differ depending on the circumstances. Given infinite resources you can build whatever you want where-ever you want. Although, if you're colonists are resource limited, then it becomes too costly/risky the deeper the colony and the bigger the rooms you want to make.
Can a settlement of at least hundreds (perhaps thousands) of people be safely built in a soft rock with low-impact methods of digging/blasting?
The safety of your rooms won't be dependent on the digging technique, any technique can be modified so that it's low-impact. When you think about blasting you need to differentiate production blasting and construction blasting. In mining, production blasting is done with the goal of getting as much stuff out as quickly as possible so it ruins the rock, in construction blasting you take a very controlled approach and you do it in such a way that you don't screw with the rocks strength. Example, Hydro Quebec has underground power houses which can be considered pretty massive: https://www.google.ca/search?q=hydro+quebec+underground+power+house&source=lnms&tbm=isch&sa=X&ved=0ahUKEwi56aSKku7JAhUBlB4KHSr0C6IQ_AUICSgD&biw=1184&bih=679#imgrc=XxoK3SQMfdINGM%3A they were made by using blasting techniques and are designed to stay there for many many many years.
Is there a way to expand such a settlement in "regular" hard rock without causing cave-ins and other hazards?
The stability of an underground opening is dependent on two aspects:
- The intact rock's strength (measured by the UCS (unaxial compressive strength))
- The quality of the rock mass (how nice the rock is, think slab of counter top granite vs gravel, it's measured using GSI, Q-System or RMR)
The stability of an underground opening will be dependent on these two values, assuming you have a VERY VERY good rock, then you're looking at being able to have a room as wide as 15m by 15m tall by infinitely long while still being on the safe side. There's a number of empirical rules have been developed over the years in order to calculate exactly how big you can make an opening. One of the most used guidelines are from the Norwegian Geotechnical Institute (NGI), it details what type of support system you'll need for a given rock quality and opening size.
Do they need to dig deep rather than shallow?
For geotechnical stability, the answer is always more shallow. The deeper you go the higher the in-situ stresses in the rock. There's a bunch of technical information that goes into this but basically, the deeper you go the smaller your excavations needs to be, too big and you risk failure within the rock itself which can lead to rock burst which can/will eventually kill your settlers.
Mind you all of this information is based on currently knowledge, if the civilization is near future then there's a good probability that would have learned more about rock mechanics. A lot of research is being done in the domain of geotechnical engineering and although we have a good enough understanding to do actual engineering, a lot of this stuff is still pretty black magicqui due to the nature of rocks. To this day, after 150 years of research predicting rock bursts is still akin to magic.