"Did" can stand alone (past simple), but "done" needs a 'helper' verb (past participle). If you hear short comments like "done already!", remember it's a condensed version of "it is already done". For using "already", it makes more sense to use present perfect ("I have already done that"), though you will hear speakers use simple past at times.

It depends on how you use the phrase in a sentence. You can say "I did that already" , "I already did that" ,"That is already done", or "That is done already." They share very similar meanings. It is all depending on "who" or "what" is the "doer" or "receiver", and what message you are trying to emphasis or get through.