No, they are completely to different items, although if your playing like 200 rounds its theoretically a masterball with high percent catch rate.
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There is not any dawn stone in Pokemon emerald, only in Diamond and Pearl. And the Pokemon that evolve from this stone don't appear in emerald/ruby/sapphire, they become available in Diamond and Pearl.
the closest you can be to a gym leader in this game is setting up a secret base with your best Pokemon then mixing records
beat the elite four and fly to lilycove city and go to the Pokemon fan club and you become a fan of the three children.
I think it makes the Pokemon that hold it become happy more easily but im not sure I also heard that it is useful to chansey as it can help chansey evolve to blissey. That's not all, the most cool is that if your Pokemon have it equipped, it gains hp when it hit opponent.
You need to get a Pineco to evolve into it at level 31. You can get a Pineco after reciving the National Pokedex at the end of the game and inserting an Emerald cartridge into your DS. They will become available at Routes 203, 204, 210, 229 and Eterna Forest.