Nature has no factor of whether or not you find a shiny Pokemon the nature will change depending on the Pokemon of course now if your catching a shiny mewthwo then everytime you catch a shiny mewtwo it should have different natures granted it's possible to have one of the mewtwo's to get that same nature of the first one or the others.
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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.
No you will always receive the same Pokemon however you may get a shiny Pokemon from the egg if your lucky.
The chance of a Pokemon received from an in-game trade being shiny is the same as the chance of capturing a shiny Pokemon in the wild.
The same way you would catch a normal Pokemon. The only difference is is that the Pokemon is shiny.
no, shiny Pokemon have the exact same stats as normal one's they are just a different colour