After you've designed the weapons and researched them, they should be on the third tab (Components, I believe) of the F5 design screen. Just scroll down the list.
Retooling is per shipyard. So Shipyard A with 5 slips gets retools to the Able class, all 5 slips can build Ables when it is done. Shipyard B with 1 slip would not be affected by any retooling changes to A.
I suspect he has researched the underlying tech systems, but not the weapons themselves.
First you research the component tech areas, which for a laser would be, for example, 12cm laser focal size, Capacitor Recharge Rate 3, and Ultraviolet laser. You don't need to research all three, you could make a laser with lower component technologies, but for this example we'll assume that you researched all three. Once you've done that, you need to R&D an actual laser by going to the research tab on the Population and Production screen and clicking on the "Design" button on the lower-middle-left. Once there you can design a laser weapons system to fit your needs.
This system is more than a little complex at first, but it allows you unparalleled flexibility to design all sorts of weapons systems, instead of relying on one or two canned, preset weapons.
Feel free to ask any question you want here, there are plenty of people here that can answer.
Kurt