Risk of fire due to wiring a new light fixture
Just a preliminary note for all diy-ers, the first thing I teach beginning electricians is the basic skill of properly splicing (wire-nutting) wires. A loose connection may seem to work for a while but it creates a lot of heat and may start a fire without ever tripping a breaker or blowing a fuse. The most important thing you should remember when you're doing your own wiring is to test all your splices. Grab each wire independently and tug on it to see if it will slip out of the wire nut newly twisted onto it. And before splicing make sure there will be plenty of copper to copper connection underneath those wire nuts between all the wires to be spliced. Do not allow any of them to back out while you are twisting. 

From your description of everything (I am answering this before any pictures have been shared), it sounds like your installation is probably safe. I say that with some reservations, of course. One thing I couldn't tell from your description is where the green ground screw is located and what you tied to it. Another question I would have for you, are the exposed copper wires at the top of the box coming from the house-wiring cables where the rest of the wires enter the box, or are they completely separate from them? If they are part of the cable assembly, they are your equipment grounding conductors and you should tie your bare wire from the fixture in with it. Even if they are separate from the other wiring, they are probably your equipment grounding system and could be tied in with the bare copper from the light fixture, but this is only a precaution to prevent shock hazards and fire if something else went wrong first. 

As for stuffing all the extra wire back into the box, don't be afraid to cut some of the extra stuff you don't need off. But I would say if you got it to all go back in there and you are certain you've made tight connections it is probably good. Things are working, you don't smell smoke or see it, that's good. Hopefully that helps alleviates any fears you may still have.