Yes you can re-use ship names. (The game will tell you that they don't recommend doing it, but it's never caused me any problems.) If you design a ship and name it, "The Best", then later make a small change to it "in game", you can use the same name and the new changes will show up in your current game.
There is an option to delete a ship design while in game using the ship designer, and that ship design will no longer appear in your build menu. You'll still have s shipconfig file in your ships folder, but it will not appear in your build menu while in game.