eli5 -- how to bring firearms stored in cases unused for 15+ years back into use

If no rust and no known issues, you should be fine just shooting them. Maybe lubricating them first, then cleaning them after. Probably the most common issue you might have would be old residue and gunk in the action or in the firing pin channel, which depending on the gun could lead to misfeeding, jams, or light primer strikes. You can attempt to inspect it yourself or just try shooting them and see how they run. There should be no real safety concerns, the only concerns would be to see if they're still running properly. If you get a click instead of a bang, keep the gun pointed downrange and count until like 10. There's a very small chance of a hangfire, particularly if you're shooting old ammo that may not have been properly stored for its entire life or if you're shooting reloaded ammo. A hangfire is when the bullet goes off a noticeable period of time after the trigger has been pulled. If you get a click instead of a bang, after waiting, drop the magazine (on the shotgun just lock the action open) and then eject the round, try to retain it and check whether the primer actually has a dent on it or not and whether that dent is similar to the dents on other rounds from that gun that did fire successfully. If you have a shallower dent, you likely have a light primer strikes. First step to try to remedy that would be to clean the gun. If there's no dent at all, either the gun failed to go into battery and needs to be cleaned or it's so gummed up that the firing pin didn't reach the primer and it needs to be cleaned.