Is the dimension "number of particles" a fundamental, or derived dimension (based on mass), or does it depend on the context, or is it dimensionless? I consider "fundamental quantities" to be those that have dimensions that are are like length, mass, time, temperature, and so on.
"Derived quantities" have dimensions that can be written in terms of the dimensions of fundamental quantities.
Let us say that I have a chemical simulation that takes into account the number of particles of a particular agent -- "moles". There is a relationship between the number of moles and the mass of the particles, so in this scenario I would say that moles are a derived quantity?
What if I am working with a model where the number of particles is a much nicer number to work with than the mass of the particles (for e.g. proteins on the order of hundreds of thousands, but not on the scale of Avagadro's number). Is the number of particles then a fundamental quantity in this context?
More specifically, let's say I am working with a model that considers forces at the level of a cell (say, forces on the order of nanonewtons being exerted by various agents on the cell membrane), while also tracking a small number of proteins. 
Would it then make sense for me to have two fundamental quantities -- one being mass in the mechanics calculations necessary for cell level effects, and the other being "number of particles" necessary for "mass law" type ODEs desribing the interaction between proteins?
 A: "Mass" and "number of particles" (henceforth "amount of substance") are in fact unrelated quantities (there's a reason SI defines both the kilogram and the mole as "fundamental units"). The reason for this is that different particles have different masses. Mass and amount of substance are as distinct form each other as mass and volume are; every substance has its own ratio (molar mass and density, respectively) between the quantities in question. 
Granted, the analogy isn't perfect - volume of a gas (or any substance really, but it's most noticeable with gasses), and with it density, depend on pressure, but amount of substance, and with it molar mass, does not depend on any easily measured thermodynamic quantity. This fact is what allowed Avogadro to discover his law (namely that, given constant pressure and temperature, the volume and amount of a gas are directly proportional) and with it the concept of the mole.
A: I wouldn't say number of particles is truly a dimension. Sometimes with a quantity like that, making a sort of dimension of it can be a helpful expedient, but I think taking it totally literally can scramble the idea of dimension. Another example of this kind of thing is luminous intensity which is just power (energy per time) weighted according to a function of wavelength that matches (as best we can) the sensitivity versus wavelength of the average human eye. I wouldn't call luminous intesity a true dimension either ... but I have seen it deemed to be one.
A: The number of particles is a conserved quantity (at large).  
The mass unit is derived from the 'number of particles'.
Take the mass unit, the kilogram  as a 'bunch of atoms' . The atomic unit is a fraction of a 'bunch of atoms'. It is a recursive definition.
Which means that, as all the 'main' equations of physics are reported to the 'electron mass', the universe can 'scale' thru time.  
from A self-similar model of the Universe unveils the nature of dark energy

Consider now mass and charge. The SI unit of mass is the mass of the international prototype of kilogram,which is proportional to the mass of elementary particles; if  their mass varies, so will the mass unit; but if that happens, one can expect that the reference atomic energy levels of the time unit will vary as well, and the time unit with them; consequently, the length unit will also vary.

