It's not necessarily a faux pas, but it could raise some eyebrows if they fly constantly, given that they're taking flight time away from squadron aircrew. It completely depends on the airwing/ship.
There's no change in the command structure when the CO/XO flies. If something critical were to come up, it would likely be handled by the remainder of the CVN's chain of command. The CO/XO aren't under CAG's command while they're flying.
It depends on their professional record up until that point. Aviators go before a major command screen board (when they're O-5s), which selects them for major command positions (CAG, Carrier CO track, Base Command, etc.). In order to be competitive at that board, they need to have been a top performer in their squadron command tour. If selected for CAG, they will first serve as the DCAG, then "fleet up" to replace CAG at the conclusion of his/her tour.
Again it depends...some officers aspire to CVN command, while others prefer the CAG route. One track is not necessarily more desirable than the other, it's largely dependent on personal preference. A CAG/DCAG will definitely fly more than a CVN CO. An aviator has better odds of making flag rank if they go the Carrier CO route vice the CAG route.