Is it true that gas particles at higher altitudes are farther away from each other than at lower altitudes, and, if this is true, does it apply to liquid particles too?

1. Yes. At higher altitudes you get lower pressures; lower pressures means that there are fewer particles for a given volume; and gas molecules move independently of each other, so they get further apart if there are fewer of them.
2. No. A liquid is in a condensed phase, which means the molecules are in contact, just like a solid. A liquid will boil at a lower temperature when the pressure is lower--because the pressure of gas above it is one of the things that keeps it liquid--but as long as it's a liquid, the molecules are in contact. When that stops, it becomes a gas.

Dan Berger