You know those things you get after beating a new mission for the fist time? It's like Sonic surrounded by a ring with wings on top and a red ribbon at the bottom? That's an emblem. (And yes there are 180 in total. You can check the ones you have by going in the "Extra" menu and then in "Emblem Results"). In the Chao world, you get an emblem for beating all the Hero races, all Dark races, all the Challenge races, all the Jewel races and I think you get one for winning all the Beginner races as well. In Karate, you get one for beating Beginner, Standard, and Expert in tournament mode. After you finish Expert, an other level of difficulty should be unlocked: Super. Beat this to get an other emblem.
Concerning the egg colors in the Chao world, the more emblems you have, the better the color selection will be. The annoying part of it is that the Black Market changes the egg colors after a certain period of time. So you might see a color you want (ex:Red) but you don't have enough rings to buy it. So you go out and get the rings, but by the time you get back, the colors have changed and the one you wanted is gone. However it will re-appear sometime sooner or later. You just have to check frequently.
The colors you named me are pretty much the basic ones, but eventually, you'll get these wicked shiny colors (Ex:Shiny Blue, Shiny Green...) I wish I could show you my killer Chao

For Mystic Melody...
Every charater has that upgrade. Right now, I can only remember the locations of Rouge's, Shadow's, Knuckles', and Tails'.
Rouge: Go in Dry Lagoon. From the start of the level, turn right and go on the ledge with a Chao box on it. (It's the ledge right behind the turtle) There should be a weird painting in the middle of the wall. It has a sort of bird, and an oval on it. Dig in the center of the oval. you'll end up in a room with a Mystic Shrine and the Mystic Melody. Get the Mystic Melody and stand next to the Mystic Shrine. In the upper-right corner, you'll notice that the action will say "Mystic Melody". Press the action button, and see what happens!
Shadow: Go in Final Chase. After checkpoint 3, you'll be at that area where you're constantly going up gravity poles (or whatever you want to call them) and being attacked by those big blue enemies. Go up on top of the 5th pole. You should then face another *hidden* gravity pole with an electric barrier that makes a full circle around it. Be careful, and jump over there. Go on top of the pole and you'll see the Mystic Melody right there waiting for you.
Knuckles: Go in Wild Canyon. From the start of the level, go in the wind thing to get you to the upper part of the stage. Go on the left side, where there are 3 "packs" of four wodden boxes (Not the side with statue's head) and go straight to the end where the lone statue is. On top of its head, there's an other one of those paintings. Dig in the center and Voilà! Mystic Melody!
Tails: Go in Hidden Base. The Mystic Melody is between checkpoint 1 and 2 (In fact it's right before checkpoint 2). Somewhere in that range, you'll be lifted by one of those red things attached to some sort of black cable (I'm sorry. I don't know how to explain it) and you'll be surrounded by those monkeys that throw bombs at you. It seems as if you can only proceed by taking a right, but in fact you can also go left. There's gonna be a type of cube with an odd picture on it (Kinda looks like a cat with wings). Shoot it three times and it should open a secret passage. From where you are, go down and go straight where there's four metal boxes. Shoot through them and the Mystic Melody will be there.
Phew....that was long. Sorry for any spelling mistakes. I know my directions are a little hard to understand, but I did my best

I'll go check where the Mystic Melody of Sonic and Eggman are and I'll post the locations later if you want, ok?