What exactly is this Lightning thing?

[Niwashi]

... and for my prize i picked the bow

You can get more than one if you want. You don't have to have the quest active in order to get the Otherworldly Package drops from the FATE, so if you want additional weapons for use with other classes, you can just do the FATE a few more times and get enough packages to trade for the other weapons.

Where is the quest for the NPC to get Snows cap? and hair or w.e it maybe. Thanks oh and what lvl do i need to be for it?

It's a level 15 quest called "Beauty Is Only Scalp Deep". You pick it up from S'dhodjbi at the Drowning Wench (Limsa Lominsa Upper Decks).