Why do rockets burst out white smoke/vapour out their sides before launching? Are they pressurizing/depressurizing or what, and why?

Because their tanks are filled with Liquid Oxygen and sometimes other such super cool propellent.  That continually "boils off" and has to be vented from the tanks when the pressure gets too high.  So the tanks will not rupture..

Initial bleed of the turbopump allowing it to start then shuts off with main engine chamber pressure when it ignites.

A failure in the sequence, the pump over speeds and KABOOM.

Ever see vids of the shuttle engines starting and notice it takes a few seconds ?,, that is the turbopump starting.

The white clouds venting before engine start are covered above.

Liquid oxygen and the like are only put into the rocket when a launch is immanent. They are very cold and are at high pressure. The bursts you see are generally puffs of liquid oxygen (or other compressed gasses) being vented to keep the pressure from getting too high as it gradually warms up.

I believe what you’re seeing is a result of cryogenic fuels & oxidizer... the shuttle would bleed hydrogen vapor as the liquid hydrogen would boil off;  Same with liquid oxygen - once it’s in the fuel tank, that tank heats it up, and it boils off, creating those vapor streams you see.