It builds the design as it exists when you click "Build" so if you change it during the building, it still produces the design when you click "Build".
No. No it does not. The characteristics of a ship are defined exclusively by the record you see in the F5 window. If you go in and change what's in that window, it will change ships already built to that design. If the ship is under construction, it will build as if it was the original design, but what comes out of the yard will be the new design. Go and try it yourself.
Actually, there's another reason to at least make the change Zook suggested. I'm sure he stumbled into this because he laid down (or built) ships to a linked design, then edited the class because it wasn't locked and he forgot he'd already built some. There are lots of good 'player protection' reasons to require designs to be locked before they can be added to the game in any fashion, be it Fast OOB, ship construction, or construction as fighter/PDC/orbital habitat.