Interesting and non-trivial questions.
Most propellants are not self-pressurized because as soon as the engines turn on the pressure would drop precipitously as the tanks quickly emptied and the propellant was unable to vaporize fast enough to keep up. LOX, RP, and H2 are the most common launch vehicle liquid propellants and none vaporize fast enough to maintain pressure. (N.B. this is at launch vehicle scale. There are self-pressurized LOX/Methane thrusters for reaction control systems.)
So in practice pressurization is performed with another gas, typically nitrogen or helium (kind of air that makes your voice funny)stored at very high pressure in a separate tank and fed into the ever-increasing ullage in the main tanks as they are emptied. Nitrogen is used because it is cheap and dense (even more dense if liquified, although I’m not aware of any such systems); helium is light.
This is a bummer in practice as the pressurization system add significant complexity and mass to a rocket.
As for mixing of helium and the propellant, they tend not to mix too much because of the vastly different densities and the large accelerations during boost. That said, special care is given to design diffusers which keep the pressurization gas from being shot into the liquid propellants like the worlds most amazing soda stream.
As for ensuring the liquid is near the outlet valve instead of the ullage gas, this is easy when the rocket is on the ground and under acceleration. It is a real problem for upper stage ignition after the main booster has shut off when the rocket is in free-fall. There are lots of solutions here, which is probably a topic for another question, but the short answer is tiny rockets to settle the propellants just before ignition, ensuring some small amount of acceleration remains to keep propellant settled, using screens or other features in the tank that take advantage of capillary action or surface tension to keep liquid near the outlet, etc.