King Mog does not have much impact in the game itself, except by introducing the concepts of the Vortex and the rest of the Farplane.
King Mog is an avid collector of things and serves as the "host" of several Events. Said events usually run for two weeks and require the player beat different levels in order to acquire exclusive Event-only currency which can be traded to King Mog for different prizes. Some prizes are universal across events while others are exclusive of each event.
Each event-currency has a time-frame in which they are valid, usually since the beginning of the event, until a few days after the event is no longer accessible. Following that time-frame that event currency can no longer be used, not even if another event involving King Mog is functional for he will request a different kind of currency, making event-currency non-exchangeable. However since event-currency are considered an item, they can be sold for Gil so the player can do this with any leftover of said currency.
These Events hold around five levels, the more difficult the level the more the currency given at the end of the level (in both total terms and also in terms of Currency/Energy). The player can also recruit additional units during this time-frame with increased odds of appearing. These units give an additional bonus to the total currency purely by bringing them to the party (even if they are KO'd). These bonuses are given in 50%, 75% and 100% fixed values for each unit (the rarest the units, the better the bonus). The player is encouraged to bring as many of these bonus units at possible since the bonuses stack (even with multiple copies of the same unit) in order to hoard as much currency as efficiently possible.
A common strategy is to bring multiple of these bonus units along with an unit that can solo the Level by itself so as to not waste too much time or resources (like Cactuars) leveling up too many bonus units. In case of difficulty the player should farm in a level they find themselves most comfortable while carrying as many bonus units as possible. This way the player should acquire a decent amount of event currency. If the player is relatively new to the game or lacks strong units, they should stick to the lower levels and if possible hoard for either Cactuars or the best equipment available. Doing so will help even the most unprepared player to slowly but surely to build up their strength.
King Mog serves as the Boss of the one year anniversary event. For INT to ELT he will do no harm to the player and will buff them, debuff himself and expel one unit once every few turns, before escaping after several rounds. An event Trial level can be unlocked in ELT level by using King's Play the first turn and then Royal Bad Mouth every turn after (both skills provided in this battle). If successful after ending the Raid (regardless of point score), the new level will be unlocked where King Mog will fight as a real Boss posing a significant challenge.