Short answers, pulled from the story.
Koichi Ishii created the original Moogle character based on a childhood drawing of a white koala with bat wings. He sketched the concept during his elementary school days before it appeared in any video game.
The Moogle was reintegrated into Final Fantasy III in 1990 after persistent requests from Koichi Ishii. The team originally placed the creatures in the caves simply because they needed someone to populate the area.
Creator Koichi Ishii suggested the name Moogle is a compound of the Japanese words for bat and mole. The English translation was handled by Ted Woolsey, who established the term Moogle for the franchise.
Yoshitaka Amano redesigned the Moogle for Final Fantasy VI in 1994. He added narrow slit-like eyes and a pom-pom-tipped antenna to the original design to create the modern look fans recognize today.
Mog is the recurring named Moogle character who originally appears as a recruitable party member in Final Fantasy VI. He has since become a fixture across the franchise appearing in titles like Final Fantasy IX and Final Fantasy XIII-2.
Good King Moggle Mog was added to Final Fantasy XIV in 2011 due to the earthquake and tsunami that occurred that year. The developers chose this character because the Tsunami and Earthquake moves they used would have been in poor taste following the natural disasters.