He is correct about the bullets. If they are fired by a robot that executes later, robots which have already executed will not see them till next round. But only if it doesn't hit them before that. Currently it's possible that bullets can be fired and will execute in the same round, traveling and damaging a unit before they have a chance to see the bullet and dodge it. Hopefully this will change in the balance update.
But execution order actually is not random, and certainly not random every round. Units execute in the order they are spawned, consistently throughout the entire game. For this reason, Archons will always execute first before other units. It's meant to feel random by design, because this makes it harder for players to determine the turn order and use it to their advantage (such as only shooting at Robots which have already executed and can't see the bullet).