Your mostly there. Yes, the carrier, base, etc must have a magazine that has has the missiles assigned as the class standard load. When the fighter is in the hanger the reload clock starts. Here's the catch for fighters, you still have to go to the F9 fighter squadron screen to actually load the missiles for the squadron. The reload clock will advance whether you've done this or not, but until you do this step the fighters will not have missiles actually ready to fire. You can also go through the F6 ship screens for each individual fighter to load the missiles, but that is an unneccessary level of micro management.
