How or when do you imagine Euron Greyjoy hired the Faceless Men to off his brother? Everyone seems to imagine it was in A Storm of Swords, but did he have the time?

Most likely, he gave them the egg before he left for the Iron Islands, then sailed, and the Faceless Man in question took the inside route against Euron’s long way around. 

Alternatively, he picked up the FM on his ship, sailed around, anchored a day’s sail or so away from Pyke, and then sent the FM in a ship’s boat to Pyke.