Usually near contemporaries, although sometimes contemporaries too. More often than not, the monarch in question doesn't get the title until sometime after their death. Alfred the Great, for example, got his epithet from 16th century authors and historians. That's typically the way it would work for most monarchs. Someone writing a biography several centuries down the road would use an epithet, and it would gradually catch on in popular use.

This is going to vary by individual. I've linked a couple of examples from the FAQ, but is there someone specific you'd like to know about?

• Alexander the Great: Was Alexander known as "The Great" even before he died? Did he refer to himselves as "The Great"?

• Suleiman the Magnificent: What feats did Suleiman I do to achive his title, Suleiman "the magnificent"?

I know about this in only a few specific contexts, but within Tibetan-Mongolian-Chinese relations, the titles (not sure if this is exactly what you mean by nicknames) were given - and often demanded - as part of the ceremony/arrangement. Many great lamas, emperors, kings, etc in the Tibetan-Mongolian-Chinese sphere have extraordinarily long names, and the length of the title is often an indicator of their fame and power. For example, the full religious names of the Dalai Lamas is often about 10-16 syllables long and might read something like "The Ocean of Wisdom, Protector of the Teaching, Moral and Compassionate Supreme Lama" or something like that and these were absolutely applied at the time of coronation/installation. The exact wording was often the subject of controversy and dispute among kings-emperors-lamas competing for power and prestige.

Interesting but perhaps not the exact "nickname" phenomenon you were talking about. Apologies if not.