Can an elementary particle be reduced to its properties? For instance, is an up quark merely its particular mass, 2/3 electrical charge and 1/2 spin? I was wondering if there was a 1:1 correspondence with a particle and its properties, but I noticed a gluon is the same as a photon in these three respects. However, my understanding is that the gluon is assumed to have no mass, so I'm not sure how much conjecture is involved. 
I'm also unsure if there are other known, general properties shared by the elementary particles where the gluon and photon differ. I thought perhaps color charge might be such a thing, but only quarks and gluons have it. Then again, the neutrinos have no electrical charge, yet electrical charge appears to be considered more or less universal, so I'm not sure what I should consider far-fetched.
 A: In realistic QFT, fields or their interaction law mostly correspond to irreducible representations of some symmetry groups. 
If we assume free theory, there is only one important symmetry: it's Poincare symmetry (it is the most important symmetry - in flat spacetime each field theory must satisfy it). Poincare symmetry leads to the statement that free particle states refers to the irreducible representation with given mass and spin (helicity). In this case we can really characterize particle only by its mass and spin. So if we set all coupling constants to zero, we really won't distinquish gluon and photon, because both of them are space-inversion invariant massless representation of Poincare group with helicity one.
When we take care about fields interaction, we usually say that there are some requirements on interaction terms in lagrangian. In general there is only one mandatory requirement: interaction terms must be Lorentz invariant. But sometimes (especially in realistic cases) interaction law is based on some symmetry group. For example, EM interaction is based on U(1) local symmetry of lagrangian, QCD is based on local $SU(3)$ symmetry etc. But for getting local symmetry we must firstly have corresponding global symmetry (for free theory in case of theories described above), which tells us that lagrangian is unchanged under sets of field transformations. Global invariance tells us that there exist corresponding operator which commutes with hamiltonian of free theory. This means that in a case of interaction the particle is characterized not only by mass and spin, but also by set of charges.
So (in my opinion) in general, if your QFT theory is based on irreducible representations of the product of groups (Poincare symmetry $\otimes $internal symmetries) then you can characterize your particle by set of numbers. Anything else will be done by the symmetry. 
A: Not really, at least not if you want to stay with properties you would normally associate to particles.
That is because particles are not the fundamental objects of quantum field theories, but fields.1 There's more to the theory than charges and masses. For every symmetry group of the theory, a field must transform in a representation of it. Now, you can associate to representations indeed a kind of charges, and often you can label them with it - but the result of transforming in a non-trivial representation of a non-Abelian symmetry is that there suddenly are more "colors" of the field, but this doesn't translate directly into a measureable property of particles.
For example, since the gluons are the gauge bosons of an $\mathrm{SU}(3)$ symmetry, and transform in the adjoint, there are eight possible independent gluon states that share everything except their "color". Yet, color is not an observable thing, since it is not gauge-invariant, so it doesn't really make sense to speak of this color as a property of a real particle - since a gauge transformation, which, by definition, does not change the anything about the physics, can turn a state of any color into a state of any other color.
Since such representation do have measureable influences on the scattering cross sections, among other things, they are a relevant property of the field. Yet they do not translate into a property of the particle other than "the particle is a state that transforms in this representation".
Also, the allowed/forbidden interactions are also not encoded in this interactions, at least not fully - you need to examine the relevant interaction terms in the Lagrangian to find out whether the Lagrangian as a whole is still invariant under all relevant symmetries. Essentially you would end up enconding all the information in the Lagrangian/Wightman functions/whatever else you think defines a QFT and call them "properties of the particle". The interaction content of the theory is not really a property of the particles, and not even of the fields, but of the theory.
You could play the word game that fields in theories with different interactions between them are different fields, but really, it's just a word game, there's nothing deep to see.

1Note the "of quantum field theories". Whether or not particles are, in some vague sense, the "fundamental objects of nature", is not the question here.
