"New Hull" just lets you add a new designation for a shiptype. This has no ingame effect other than the labeling. You could create an asteroid miner called the "Infinite Tyranny class Genocide Platform" if you want.
If the refit cost for a ship is under 20% of the total shipcost you can build/refit it out of the same yard. In practice, only pretty similar ships will be able to refitted. There is also an extra overhead if there is a tonnage change. Note that refits don't always go under 20% both ways - you may be able to build Ship A and Ship B out of a yard tooled for Ship A, but only Ship B out of a yard tooled for Ship B.
Ideally ships that you want to be interchangeable have similar roles and have identical non-mission components - the same amount of armor, engines, fuel, crew, and engineering spaces. The easiest ships to make interchangeable are AMM and ASM missile ships as a lot of their cost is in magazines rather than the launchers themselves. Regardless, keeping ships on similar chassis helps keep down retooling times and costs, as that is also dependent on refit cost. So even if you can't build out of the same yard there are some advantages.
There are some other tricks to building interchangeable ships. Mainly you want to keep the 'mission payload' to as small a portion of the ship as practical. Fast well defended ships are good candidates - it puts a lot of tonnage/cost into engines/armor/shields. If at all possible, you want to standardize your electronics as well. But that is tough.
You can view the refit costs in the components/DAC tab of the ship designer.
Personally, I fiddled with retoolable designs for a while but found I liked multirole ships better.