Is it permissible to perform the remembrances (dhikr) of the Adhan in cases of waking up after it, or hearing it from the middle, or not hearing it while knowing its time has begun, or hearing it while relieving oneself?

Whoever does not hear the entire adhan or part of it due to sleep or distance, is not required to repeat it nor to repeat what they missed; because the time for its requirement has passed. It is apparent that responding to the muezzin is contingent upon hearing.

As for whoever is relieving themselves, they are not required to repeat the adhan, neither verbally nor inwardly. If they finish and there is still some of it remaining, they may follow along.