Learn Turkish: Bugün dışarı çıkmıyoruz, hava zaten kasvetli. - We are not going out today; the weather is gloomy anyway.

It acts as a reinforcer, similar to "anyway" or "besides" in English. By saying hava zaten kasvetli, you are presenting the gloomy weather as an undeniable, pre-existing fact that perfectly justifies the decision made in the first half of the sentence.

Is it possible to translate zaten as "already" in this sentence?

Not exactly. While zaten often translates to "already" (as in "I already did that"), here it is functioning as a discourse marker. It introduces a supporting argument. Translating it as "The weather is already gloomy" misses the point; "anyway" captures the intended vibe much better.

Why doesn't dışarı take the dative case (as in dışarıya çıkmıyoruz)?

When directional words like dışarı (outside), içeri (inside), or aşağı (down) are placed immediately before a verb of motion like çıkmak (to exit), they usually drop the dative suffix in natural Turkish. Adding the suffix is not wrong, but leaving it off is much more common.

Why is the present continuous tense (çıkmıyoruz) used when talking about the rest of the day?

Just as in English ("We are not going out today"), the Turkish present continuous tense (-iyor) is frequently used for planned events or firm decisions in the near future. It shows the decision is already made and is currently in effect.