وعليكم السلام ورحمة الله
* already= تم -قد - بالفعل
*as you said in your example ( I did that already= لقد فعلت ذلك مسبقاً / لقد قمت بذلك بالفعل)
*Not yet = ليس بعد
*ٍStill =(مازال / لا زال) the good question here is the difference between the 2 words
* لا used for negation when the verb is present like لا يزال it means still ,but when لا used before past it means wish like
( لا زالت صحتك جيدة = i wish you a good health)
*ما used for negation when the verb is present or past but also there is difference when you use it in present it means negation forever (ما يزال الله رحيماً = the god is still merciful forever) ,but when you use it with the past it means negation till a certain period ( ما زال الطالب في الاختبار = the student still in the exam "it means that he will finish it after a certain period") .
* I'm still doing that =لا أزال أقوم بذلك / ما زلت أقوم بذلك