1. yes, you can store modded modules
2. yes, you can interchange modules between ships, you just need to remember that they are different sizes, so if you have a modded class 4 fsd, it won't perform so well if you install it in a ship that usually carries a larger one (if at all) and it won't fit at all in a ship that only has space for a smaller sized FSD. Same thing occurs with some* of the core modules, you can downsize (commonly done to increase jumprange) but you will have trade-offs like being unable to boost etc. afaik the only ship-specific module is bulkheads - which you can still swap between two of the same ship designs, but you can't swap between different designs
*not playing atm, but from memory powerplant, thrusters, FSD, Power distributor, Fuel tank can be down-size. Life support, sensors and bulkheads can't