In the British navy during the period I study, gunpowder on ships was kept in a special hold (a "magazine") that was isolated from the rest of the ship. The magazine was entered through a short passage that had an all-wooden outer door and an inner door that could either be all wood or sometimes hanging cloth that could be dampened. It was guarded by a Marine sentry at all times. It was lit by a lantern in a separate compartment, behind glass, and it generally had a small "filling-room" that was entered before the main magazine where the gunner or his mates could work at filling cartridges. They would wear "list" slippers, or soft slippers, into and out of the magazine and get rid of any metal implements before entering. All these precautions, of course, were meant to avoid sparks or flame reaching the powder barrels.

but would the waterproofing have been sufficient for the contents to remain dry if the barrel fell overboard and floated in sea-water for a short period of time or would the powder immediately become unusable?

The barrels were generally sealed tightly enough that a mild dousing or dunking wouldn't harm the powder. If powder did get wet or damp, which was fairly typical on board ship (wooden ships leaked constantly and dampness was a fact of life between decks), the gunner or his mates could re-corn the damp paste and lay it out to dry in fine weather (gunpowder of the era was made by wet grinding).