I'd agree that as many of those extra mounts as possible be moved off of your action bars to make way for the essential skills. I'm not sure how many heavily-used combat skills mages have, but as a rogue I can get by with 2 1/2, 3 tops if I throw in some situational macros. That would give you a lot...
May want to consider reducing the chat window size or getting a chat addon that can do such. It's taking up a good amount of real estate.
Also might be helpful to post an image during combat to see what else is popping up and taking space (i.e. any WeakAuras, timer bars, combat text, etc.).