You get the quest then go do the FATE. The FATE that had Lightning may not spawn Lightning again the next time around.

Upon getting gold you get 2 Parcels, this works like the Bomb Ash meaning you never need to even get the quest.
To turn in the quest you need to give 1 Parcel away, and next to the quest giver is an NPC that sells each of the weapons for 1 parcel.

You can then go back and farm more parcels and get each weapon for every class.

You do not need to do the level 45+ quests on a MNK to get the MNK weapon, for example.

The quests are for the armor that you get, which can be worn by any class. So as long as you have one class over level 45, you'll be fine.