Well, I'm not entirely sure what the exact odds for getting a shiny Pokemon is, but from what I recall, it was 1 in 8192. And even if you spent months trying to get it, you might not get it. If you really want that lotad though your best bet is through chaining with a pokeradar in Pokemon pearl, diamond or platinum with sapphire plugged in. Or through the Masuda method. (search online for further detail)
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safari zone
You can get lotad in the safari zone if you place the right items. to find out exactly how look at the pokemon marriland or serebii safari zone guide.
I believe you can get a Shiny Stone in HeartGold in the Pokèathlon Prize Corner after you get 5000 Points.
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 can buy the Shiny stone at the pokeatholon dome