Use the Cryomancer now or waste 3 mana and wait for +2/+2?

In this case specifically, there's an empty board, you have three other 6 mana cards, and if you don't play Cryomancer, you have no play. In this case, taking the tempo and playing the 4/4 is incredibly valuable. In situations like this, you should absolutely go for tempo over value. However, if you had other five mana plays, throwing one of those down and playing the cryomancer on turn six would generally be preferred.

Playing Cryomancer with no buff on an empty board would have absolutely been a waste. If your opponent has no pressure on you, there is no point in wasting the +2/+2 opportunity, especially since you know you can use it next turn.

There isn't actually one answer for all cases here. You already know what deck you're playing, and by turn five you should have a good idea of the type of deck your opponent is playing (face, tempo, combo, control).

You already have a turn six play in the archmage, so in all matchups except one, you want to play the 5/5 on turn five for the tempo (this statline is only slightly below vanilla stats for a five mana card and totally fine to play tempo).

The only matchup where you would want to wait is control-vs-control where your deck is slower/heavier. In all other cases the tempo from the 5/5 will be much more useful to the coming turns.