Greencheeks are awesome little birds. I've had one for about 5 years and she rules the roost.
Nippy is a general greencheek trait. Sure, there are exceptions to that (mostly babies..), but as a rule they're pinchy little birds with very subtle warnings. Often it's beak first, figure out what you did to cheese me off later.
They are quieter than Aratinga conures like suns and jendays, but can still belt one out when they want to.
As for the cage aggressive thing, we do the same with Kizzy that you are doing. She willingly scurries out of her cage onto the open door, so we luckily don't have to stick hands into the shark cage. Just be patient and forgive him his crabbiness when he has his moments.
They love toys, and are little busybodies. Soft wood like balsa is a favorite here. As above, a pelleted diet supplemented by fresh foods and a few treats is healthiest. Zupreem isn't the best pellet out there because of the sugar and added coloring, but it's better than no pellet at all and is generally loved by most birds above the natural pellets. I have a 10 pound bag of Zupreem fruity in a size that'd be good for your guy. If you're comfortable sending your address to me via PM, I will ship it out to you free of charge for you to try. For fresh foods, the seven layer salad is awesome and contains a bit of everything.
http://www.aviannetwork.com/articles/salad.htm
My birds love rooting through it. Sprouts are wonderful too, but not everyone has the time to do those.