If it was obtained legally, then yes.
Off the top of my head, I think Gen 1 wild grass/cave/surfing encounters will never give shiny DVs via how the RNG works. Only gifts, trades, static and fishing will be able to.
From Gen 2 to Gen 1, yes.
It just won't appear shiny in Gen 1, but you trade it back to Gen 2, and boom, it's shiny again.
Gen 2 uses DVs to determine whether a mon is shiny, and there is no legit way in-game to manipulate DVs of Pokemon you own.