Thought it might be nice to have a thread for answers to the moogle tests. I only got to try a few tests in Sandy before work: art, history, architecture, and astronomy.
To start these tests you have to get a moogle guard from a red chest next to a moogle (in Sandy it was underneath the tree behind the circus in South Sandy). Once you have the moogle guard equipped you can start talking to the proctor moogles.
For the architecture test (close to the gate to e. ron) you have to differentiate between several building fronts: The Lion's Spring Tavern, Chocobo Stables, Cathedral Reliquary, Cathedral Manuscript Room, and a noble house (think it's Count Caffaule's Manor).
For the art test (near the tavern) you have to identify NPCs who are not part of a guild (my time was carpenter's guild).
The history test (across from the chocobo stable) is about Gwynham Ironheart's journey. You can answer it using
http://wiki.bluegartr.com/bg/Stone_Monument.
The astronomy test (near the tavern) shows in game constellations (someone said it's from a windy mission). I have no clue how to identify them and didn't find any information from any wiki.
Any more information on the tests (especially the astronomy one) and if there's a reward at the end of answering all the proctor moogles would be greatly appreciated.