From this, I basically just pull colors, and do my best lol. Is there something missing from this?
If you're really into color theory:
http://www.huevaluechroma.com/I wrote and rewrote a lot of different explanations for why your colors are "wrong" and "don't work", but... none of them were valid.
It may just be lack of ramps. Arne's 16 color palette is great because the colors can fade darker into each other. This also creates some hue shifting harmony.
That's all I got. Hopefully crab2selout.png and others who know more about color theory here can help.
As for the NES reference guide, is there a book or a website or something on what the specs for NES art is?
KittenMaster already linked the huge post I wrote. There may be some inaccuracies, but I did my best to include only stuff that would matter to someone making NES graphics. If you have any questions, or want any of that explained in a different way, post about it.
Max number of frames per sprite
The console doesn't limit you in this way. Everything else (tile limits mainly) affects how you decide how long your animations should be. Unless you mean something different by frames.
Max size of a sprite
Jerk answer: 8x16. That's as big as a hardware sprite can be. You get 64 of them. So...
if you laid them all side by side: 512x16
Top to bottom: 8x1024
In an 8x8 arrangement with no overlap: 64x128.
But that means you're using all your hardware sprites to form one "sprite". You could show one sprite this big every other frame and make a flickery fighting game. In short, the answer is like the above. The other limitations are what would affect how you would think about this. You could show a "sprite" bigger than screen if you didn't mind super sprite flicker.
I make NES games, but I think the NES palette is really not that great. There's not a real yellow, and 7 of the colors are wasted by being black. Like crab2selout.png said, the colors don't blend.
Honestly, I'm totally messed up about NES stuff. Seeing "NES style" makes my blood pressure rise because I actually have to check to see if it really is. And even if graphically it is, gameplay wise it usually isn't. So many things people make happen in NES style games would make the NES processor slow to a crawl. There's more to authenticity than graphical authenticity.
But I'm a purist, and that's off topic.
Here's some interesting NES art:
http://jamesmorrow07.deviantart.com/gallery/11869769As far as I can see it follows restrictions, but I didn't get out my magnifying glass.

If I can find more sources of some of the stuff I have saved, I might link more.