Is one allowed to use the cold water from a water cooler on Shabbos when it is plugged in?
A water cooler is essentially a fridge. Still, even with a fridge, the usage of modern fridges on Shabbos isn’t that simple. See here for more details:
Halacha2go.com article #700: Hot Debate: Keeping Food Cold on Shabbos
Furthermore, a cooler is worse, as the effect is more instantaneous.
In the case of a cooler, one should neutralize the UV light before Shabbos, if there is one.
It is also recommended to disconnect the electricity before Shabbos and check if water still comes out. If no water comes out, this proves that the operation is electrical and not mechanical. If water does come out, it might be only mechanical. Therefore, it would be best to use it only while the electric is disconnected.
Some suggest to take water only when the motor is operating or to take only up to one cup at a time, an amount where there is no certainty that the system will begin operating again.
To avoid the issue of Borer, it is recommended to let the water run a bit before using it. See here for more on this:
Regarding water left in metal overnight, see here:
דנו בזה מצד מלאכת בורר, הפעלת חשמל, וכן מצד כותב – באם מופיעים שינויים בצג הדגיטלי.
וראה מטבח כהלכה ע׳ 221 ואילך. מקור מים חיים מסילתי ע׳ 65 ואילך.