After you beat the elite 4 go talk to the old man in Pewter City. He will give you the silver wing. Then, go to the Whirl Islands from Canalave City. To get there, you need surf and whirlpool. Go through the whirlpool and sail to the cave. Enter it. You do NOT need flash to navigate through the cave. Although it will help you, I got through the cave without it. When you come to the Japanese man, go up to him and press a. The silver wing will shine above his head. Then, he will let you through. Go through the zigzag path and you will find Lugia. To get shiny Lugia, before you battle him, save the game. Battle him. If he is not shiny, (the blue spots will be red and he will sparkle) do a software reset. Press L,R,START and SELECT simultaneously. Keep on doing this until Lugia turns red and sparkles. Don't give up though! It takes a long time. I am still trying to get one right now. The odds of getting a shiny are 1/8,192. It will take a lot of trying (and turning your fingers to bones from all that soft resetting) but it will be worth it in the end. SHINY LUGIA IS SOOOOOOO EPIC!
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Yes the same way you get shiny Pokemon in the other Pokemon games is how you get shiny Pokemon in heartgold. There is a free shiny Pokemon in the lake of rage.
You need Ho-oh and Lugia
Luck. You have a 1 in 8,192 chance of your starter Pokemon to be Shiny.
I believe you can get a Shiny Stone in HeartGold in the Pokèathlon Prize Corner after you get 5000 Points.
To get any Shiny Pokemon, you need: LUCK