No.
After around 3,543 resets, the dragon finally shined. The only visual difference is the sparkles, nothing else, and obviously you can't catch it. As a fun test, I sent out a Ditto, and it transformed into a shiny Origin Giratina.
As for my second question, when you view it in the Pokedex, the Origin form will appear, and it is shiny.
Late edit: I realized I never added images as proof, so I'm adding them now.
The shiny shadow Giratina, after around 3543 resets.
Ditto transforming into a shiny Origin Giratina.
Shiny Origin Giratina showing up in the Pokedex (I didn't get the Griseous Orb).