Does the game roll for my dodge chance when the enemy weapon is fired, or when it would hit my ship?
Dodge chance takes effect differently depending on whether or not you have shields.
If you have no shields, dodge is calculated when your ship is about to be hit.
If you do have shields, dodge is calculated when they are about to be hit.
This may seem like a trivial distinction, but there is one important result of this: Missiles. Even though missiles bypass shields, the dodge is still calculated as the missile passes the shield barrier.
So if you are using a Cloaking Device, use it before things hit your shield.
It seems to be when the shots are about to hit your ship. I use my cloak right after the enemy fires its volley which takes my evade to around 90%-105%, giving a hit free volley cycle for me every time.