It is Lugia that has been corrupted by Cipher, so it is evil now. You may purify it or turn it back to normal.
Only in Pokemon XD: Gale of Darkness. You can not transfer it out of the game as Shadow Lugia, only as normal Lugia.
It is a side-game. If you mean is it fan-made or something, no, it is real.
Shadow Lugia (Japanese: ダークルギア Dark Lugia), codenamed XD001, is a Lugia corrupted by the criminal organization Cipher. It is believed to be the ultimate Shadow Pokémon, completely immune to purification; to reflect this, the "XD" in its codename stands for "Extra Darkness". As a sign of its corruption, unlike other Shadow Pokémon, Lugia's appearance changes from that of a normal Lugia. Shadow Lugia has only been seen in Pokémon XD: Gale of Darkness.
I've owned this game before and have played it many times.
Credit to Dark Goku for link to picture and answer 4.