Gifts for Discerning Monsters
It appears that no one really completed a guide for gifting Monsters in Guardians of Azuma. People wrote down the lists of 'drops' or products they obtain, but did not pursue the actual lists in any comprehensive fashion.
Part of the problem is, in my experience anyway, that every Monster is unique. A gift that works to send one King Wooly to your barn may fail with the next.
Be that as it may, I now will give my results here:
Silver Steed on Island: tamed with Strong Fur
King Wooly: tamed with Medium Fur
Iron Tengu on Glacial Island: Quality Fletch
Moon Mushsprite: Heavenly Kingdom: Sleep Powder
Golden Gatling in Heqvenly Kingdom: Olive Oil
Grizzled Bear Frostfang took Quality Ckaw
Buffsmoo Frosrtfang took Strong Horn
Red Oni Archer Fleeting Flower took Strong Cloth
Spider took Quality Thread
Mystic Neko took Quality Cloth
Goldblossom Mine:
Blue Oni guard rejected Quality Horn but took Quality Cloth
Underworld:
Lantern Spectre: Spirit Cluster, rejected Quality Fang
High Level members of the Wooly family give both Medium Fur and Large Fur. Working on advancing the Skill Nodes that offer a higher level trophy randomly and equipping an Accessory that gives a better chance to receive rare items are ways to boost Large Fur offerings.
Also it is useful to know where to find the high level monsters that feature in advanced Requests:
Ruffamoo Level 160 found in FoE Level 9, as are Onis, Oni Archers, Colossal Onis, and a Horse.
Lanterns and Parasols found in FoE Level 5.
Lantern accepted Spirit Cluster. Parasol accepted
Mystic Nekomata found in Level 7.