the sun - If there are neutron stars, would most stars be considered "proton stars"?

Protium is a proton + an electron.

Under enormously high pressure, it's energetically favorable for electrons to merge with protons and become neutrons - see here.

are stars mostly protons

By mass, yes, at least before they get too old.

The mass of the universe is more complicated, but anything solid that we think of as matter is made of atoms, which are by mass, mostly protons and neutrons (you can break it up further than that if you like, but that's best for another question).

Most hydrogen has no neutrons so any hydrogen rich object (the sun, most young stars, gas giant planets) are by mass, mostly protons. That's no longer true when a star gets close to the end of it's life and has burned much of it's hydrogen.

Jupiter, by mass is roughly about 80% protons. The sun, because it's been turning hydrogen into helium for about 4.5 billion years, is roughly about 67% protons by mass. The Earth, mostly other elements, Oxygen, Silicon, Iron, etc, is a about 50% proton by mass.

Would stars like our Sun be considered "proton stars"?

I suppose you could use that term, but I don't see any benefit to it. It's not too different than calling the sun a "hydrogen" star. All stars start out as hydrogen stars.