I doubt that, why would they? The issue with not letting the FAA get Spitfires was production, rather than in service aircraft. The admirals reasoned that if another manufacturer, i.e. Fairey built them, then RAF number isn't compromised. By August and September 1940 fighter production was matching pre BoB numbers of fighters and by the end of October the RAF had more fighters in service than what it began the BoB with, so it compensated for its losses and added to its overall number. This meant the calls for stop-gap fighters went unanswered simply because they weren't needed, so if the FAA had its own single seaters, I'm pretty sure they'd probably assist the RAF and base them at RAF airfields, as the FAA did with its torpedo bomber units. The Swordfish that attacked the Scharnhorst and Gneisenau during the Channel Dash were temporarily shifted to RAF Manston.
Probably not, it depends on whose building it. The Barracuda was a dog's breakfast aerodynamically and that was the result of combining the two requirements. As mentioned, dive bombers don't need to be big. Torpedo bombers do. The bigger the dive bomber the greater the amount of structural strengthening, which adds weight and complexity in undercarriage and dive brake design etc, which requires more power to get the heavier frame off the carrier deck, so a bigger, thirstier engine, which reduces fuel consumption, thus reducing usable range etc, etc... An extreme example is the He 177 as a dive bomber and the issues that had.
Let's put it this way, in my opinion, the Skua is not worth modifying and history proves me right on this, it was retired after only a few years in service and it wasn't modified beyond the basic design in multiple variants. The alternative is to create something else within an appropriate time period.