I like the direction you are going, because it really does create it's own niche for the weapons.
That said, we have historical records of people with a brace of pistols, firing one then grabbing the next and the next. I think the short/long rest requirement sort of forces that sort of play on anyone who is willing to invest in the feats (or proficiency if they don't have it). I think this may be too conceptually limiting, and just purchasing more (assuming they are purchasable) and restocking loaded ones from the ubiquitous bag of holding between encounters gets around the rest requirement as a limitation. I'd make it a simple 1 or 5 minute process. You're already breaking convention to do something interesting, make them effectively "per encounter" if you want.
Damage looks good to me, because you are not adding an ability score to it.
Firing a pistol in each hand is a genre staple in genres with with firearms, I'd think of a way to allow that. The goal isn't just to be balanced, or even interesting, it needs to be those and also support the feel/theme/genre you are going for.
I might overload some other feats instead of creating just new ones. Either by specifying the type of action it is so it overlaps, or just adding to certain feats. (BTW, was this intended to be used with 2014 or 2024?)