1. Yes, if your opoonent got the first turn, then you use them at the start of their turn. You use them at the start of the entire round.
2. Yes, using it in your turn if you went second is too late.
3. No you can't. Before the game is the only time described as being out of phases. There is no space between turns which does not count as a phase, so "Start Of THe Battle Round" would be at the start of a movement phase.
4. All Stratagems are once per phase only.