yes, they won't do you (so to speak) too early in life, as your eyes are still changing.
The laser will allow for any slight eyeball movement, so if you flicker, the laser will move according to where your eyeball is.
They use a clamp to hold your eye lids open, and I think it might somehow also very gently hold your eyeball in place, but I'm not sure. I only think this as I had a bruised right eyeball, and when I asked what the pain was, was told that the clamp on my eye had bruised me, and that it'd go away like any other bruise. I hadn't quite realised up until that point that the clamp was also on my eyeball. Maybe it's better that I didn't know that at the time! You also stare at a fuzzy orange light, which is really easy. It's easier to stare at a fuzzy light than a clear one.
The only anasthetic you get is drops into the eye. You are fully awake and alert through the whole thing - all 2 minutes of it!