Why are particulate matter measurements (like PM2.5) from an air quality monitor given in micrograms per cubic meter (μg / mm^3)?
The fundamental measurement for particulate matter is done using a filter. You pull in a known volume of air over a known period of time, and then you measure the mass of the filter before and afterwards. This allows you to calculate mass per volume (on whatever time average you choose). This method is the most accurate way to measure particulate matter and is the federal reference method for determining attainment of federal air quality PM standards.
I assume the device is using an optical system to count particles in a sample volume the air, and extrapolating as necessary to a standard volume like a cubic meter.
While these devices do exist (particle counters like purple air sensors) they don't measure mass directly and are incredibly dependent on the mass per particle assumptions. There are also light scattering nephelometers that are more accurate than a particle counter, but they don't measure mass directly either.
If the device does indeed count particles, how does it convert that to mass? Is the mass of 2.5 micron particles suspended in air consistent enough that multiplying the count by the average mass yields a good estimate?
No it's not consistent enough, and the inference is incredibly dependent on the type of PM you assume you are measuring. For instance, if you are measuring 1 million dust particles, it will be far heavier than 1 million smoke particles, and the particle counter won't know the difference. But, they do it anyway. For a nephelometer, best practice is to co-locate a nephelometer with a filter-based monitor so that you can build a correlation relationship that is appropriate for the PM sources in that region. However, when PM emission source profiles change (e.g. a wildfire smoke event becomes more prominent than local sources) the correlation is no longer valid and thus reported concentrations are erroneous.
Is a correction factor for air pressure applied to account for the fact that heavier particles are more likely to settle out of suspension when the air pressure drops? (My particular monitor does not report air pressure, but I suppose it might still monitor it.)
Air pressure doesn't change due to suspended particulate matter, but air pressure is an important variable because in order to report consistent data you need to account for temperature and pressure to standardize your air volume to STP (or similar).
Is the health concern of breathing small particulates in a particular size category based on the total mass of them inhaled or the quantity of them?
The smaller the particles, the deeper they go into your lungs when you breathe them. So, for instance, ultrafine particles (PM1.0) are a bigger health concern than fine particles (PM2.5), which are more of a health concern than coarse particles (e.g. PM10 - minus PM2.5). All health based standards for particulate matter are based on mass concentrations.
Intuitively, I would have expected the measurements to be presented in parts per million or as a count of particles per volume, not only because of how (I assume) it's measured, but also because of the utility.
This is how gaseous measurements are reported. Individual gases do not have different sizes because they represent a certain molecules. Particulate matter can be many different molecules with different sizes. Furthermore, particulate matter is in solid or liquid form, while gases are not. And, gases are not measured using filters (which can be weighed) because they do not collect onto a filter.