i dont know if it has been posted but anyway
to get a shiny legendary,just save in front of the pokemon
then battle if it isnt shiny push a+b+start+select
keep on untill its shiny
it works!

yes, the stats for stationary legendaries are set when the battle begins (for the legendary dogs, they're set when you fix Celio's machine)

It takes an awfully long time, though - 1/8192 chance

Do you have to cut the game off at any time during that?

I think if you want a shiny legendary dog, you have to cut the game off right before you give Celio the Sapphire.

However, it's not that useful - you have to find the legendary dog again each time.

For other legendaries, you can reset at any time during/after the battle (you should save right before the battle with the legendary)

Another thing, if you Reset, And its Shiny, Dont Reset again, because it wont be. When i caught my Shiny Uxie, i Restarted it at least 10 times because i kept Killing it.

Thank you for this totally useless information!