Identification of particles and anti-particles
Good question. It's nice to see somebody thinking about physics. Shame it's an old question, but hey ho, it's never too late for physics.
The identification of an electron as a particle and the positron as an antiparticle is a matter of convention. We see lots of electrons around us so they become the normal particle and the rare and unusual positrons become the antiparticle.
Yes. It's a matter of convention. Not a matter of particle properties. Think of a 2 x 2 table and list the properties of the electron, the positron, the antiproton, and the proton.
My question is, when you have made the choice of the electron and positron as particle and anti-particle does this automatically identify every other particle (every other fermion?) as normal or anti?
No. Not unless you do physics "by convention" instead of looking at the hard scientific evidence.
For example the proton is a particle, or rather the quarks inside are.
Forget about the quarks. We've never seen a free quark:
Image credit CSIRO, see The Big Bang & the Standard Model of the Universe
Let's stick with hard scientific evidence. Let's focus on the four stable massive particles. One of which is the proton.
By considering the interactions of an electron with a quark inside a proton can we find something, e.g. a conserved quantity, that naturally identifies that quark as a particle rather than an antiparticle?
Or do we also just have to extend our convention so say that a proton is a particle rather than an antiparticle?
No we don't have to. Of course, we could, and then we could wail about the mystery of the missing antimatter. We could sweep lepton asymmetry under the rug and marvel at baryon asymmetry, and put it on the cover of magazines.
To complete the family I guess the same question would apply to the neutrinos.
Forget the neutrino. People classify the neutrino as a lepton "by convention" instead of looking at the hard scientific evidence. The neutrino moves at c or so close to c that you can't tell the difference. Its mass is so close to zero that you can't tell the difference, and so is its charge. Let's see now, what particle does that remind you of? Ah, the electron!
As regards the thrust of your question, see this article about positronium which says "to a first approximation it can be regarded as a sort of light hydrogen atom". Positronium is a short-lived exotic atom, comprised of an electron and a positron. We call the electron matter, and we call the positron antimatter. And positronium is like light hydrogen. So why do we call hydrogen matter? Throw away that convention and you could justifiably say the proton is the antimatter and the antiproton is the matter, and then say hydrogen is an exotic atom too. Especially since baryon asymmetry is counterbalanced by lepton asymmetry. I rather think it's something like a game of mixed doubles in tennis. One side was always going to win, and did win. But then we called the winning team the matter. And now we wonder why the ladies lost, when actually, they didn't: