There are atoms without neutrons, but protons and electrons are both universal to all atoms, right?

Pretty close. Other than hydrogen (no neutrons) and its isotope deuterium (one neutron), every stable atom has neutrons (plural). Every other atom has two or more neutrons -- and except for a fairly rare isotope of helium, at least as many neutrons as protons.

Depending what you mean by "usually" the last part of the statement may or may not be true. While virtually all kinds of atoms (other than hydrogen) have neutrons, the great majority of atoms in the universe (being hydrogen) don't.

Of all the joys in life, I think I like nit-picking the best. 🙂

H+ is just a proton - its more of a perspective thing. Chemists think in terms of elements and physicists think more in terms of fundamental particles.

Yes.