Would the atmosphere cease to exist?
Both temperature and pressure dictate how the atmosphere precipitates (falls down).
Currently, on earth, sometimes the gas will fall down as liquid. You know this a rain.
In other planets, it can precipitate as liquid or even solid. Some planets rain down iron, some even diamonds.
The atmosphere will generally exist, but this is based more on gravity than temperature or pressure.
Can the whole thing “rain down” though?
And can it “create itself” again (go back up afterwards)?
The entire atmosphere as a whole? Probably not as that would require absolutely massive amounts of energy to do on a short time scale.
The moisture portion of the atmosphere does condense, fall as rain or snow, and then evaporate back into the atmosphere on a regular basis. Other parts of the atmosphere does condense or expand with temperature and that is why we have high and low pressure systems even if they don’t go all the way to pure liquid.
So the things you describe happen to some extent, but the entire atmosphere liquidating and condensing would be unlikely to happen for long as that amount of energetic change is going to involve some crazy amount of energy to change back and forth.