Completely with you on this one. There's just no logical in-universe reason for the warp behaving the way it does. Now, screamingmetal does have a point - there is the little wrinkle of transwarp beaming that was supposedly Scotty of the future's idea (even though Scotty hated the transwarp project in STIII). Scotty does mention in ID that Starfleet commandeered his transwarp equation, and it obviously was used by Harrison. Maybe all starships were outfitted with it? That would obviously change things.
Now, as far as the Vengeance being faster - there is an in-universe explanation. In the original timeline, They "rewrote the warp scales" somewhere between TOS and TNG - and the TOS ships could only go a max of warp 7 or something like that, while TNG could go warp 9. So it could be that the Vengeance tech is more akin to the TNG scales than the TOS scales.
As far as the Vengeance being able to fire weapons: Phasers only travel the speed of light (at least, in the original universe). Torpedoes could in theory have mini warp drives...
If you look at warp theory, theoretically there's a bending of space around the ship, creating a bubble of normal time around the ship but shifting space around it (hence the name "warp"). The only way for this to make sense would be if somehow the Vengeance entered the Enterprise's warp bubble, then the space between them would've been normal space-time - but we've seen before (I think it was in Enterprise?) that's it's crazy difficult to have two ships in the same warp bubble even if they both WANT to be there. so the Vengeance would have to have an impossibly good pilot/computer, or it has some new tech that didn't exist in the old universe.