I don't really think there is a set time that your king starts to move around. Generally the best time to bring your king in to play is in th endgame, as you described, to support your pawns and minor pieces but you should act accordingly. It really depends on the endgame and the material that each side has. If your opponent has a few minor pieces but you only have your king and your pawns, then you should look for a way to promote your pawns and see if you king and help without putting him in danger (although at that point I would personally resign since I don't have the endgame skills).
In short, I personally wouldn't bring my king out if my pieces can still play efficiently but if can use my king without putting him in danger I will. But, hey, I'm no grandmaster.
When the queens are gone but rooks and minor pieces are still in play should the King remain hiding like a frightened rabbit or become more poised to become a fighting monarch?