Is there any difference between all these "already" words, and is there among them a "universal" one which can be used in any context?
The answer would be "もう"
I'm not sure how many contexts it could be for the usage of "already" but here's just a few basic examples:
I already ate = もうご飯食べた
Oh, it's already 12? = あれ、もう12時ですか
We're already late = 僕ら、もう遅れてるんだ
I already know that = もう知ってるんだよ
Hope it helps.