The biggest problem with using established side characters from the show (Dukat, Garak) is the need to explain them. There will always be people going to a Trek movie just for shits and giggles, people who have never seen TNG for example. When you then do a movie about Q, they're going to wonder just who that is, without using half the movie to 'explain' Q to non-Trekkies.
Not really. I don't think one needs to have seen Space Seed to enjoy TWOK or Best of Both Worlds to enjoy FC (and of course the Borg Queen didn't even appear in BOBW).
Characters like Carol Marcus were also supposed to have had a history with Kirk but, even though they'd never appeared in a tv show or movie before, their connections were generally established with a line or two of dialogue. Why would it be harder or more distracting to do so with a character who has appeared onscreen before?
Good point. We probably overestimate the difficulty of reviving old characters sometimes, forgetting that it's not necessary for the audience to know every tiny detail of an old character's backstory just to follow the story on the screen. After all, The Wrath of Khan
didn't need to mention poor Marla by name, let alone explain that Khan's dead wife was a former Stafleet historian . . . .