I am pretty sure this works the same way as other shiny Pokemon (approx. 1 out of a million chance.) I don't knnow for sure. However you do get to catch Mewtwo, i know because i did using only (surprisingly) 2 ultra balls(<sounds wrong<)
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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.
you go to the third gym you go up you go to the grass and you randomly walk around and voila! dark lugia if you want shiny do the same thing but release your mew and mewtwo and there is shiny lugia.
By cheating.
1/8192 is the appearance rate for any shiny pokemon. The odds are no different with legendary pokemon.
The same way you would catch a normal Pokemon. The only difference is is that the Pokemon is shiny.