As far as I can tell, from the top left, counterclockwise the requests are: The Jeweled Hourai Branch, the Dragon's Jewel, the Robe of the Fire Rat, the Buddha's Stone Bowl and the Swallow's Cowrie Shell.
While normally barren, in the presence of impurity it sprouts beautiful seven-coloured jewels, allowing it to act as a detection system.
The Jeweled Branch of Hourai is the branch of an udonge which has been brought to Earth; the Lunarians use these to sow conflict on Earth, since its beauty will grow in accordance to the impurity of its holder who will then take it as a sign of their authority.
As I see it, the branch itself has no direct effect, it just facilitates corresponding behaviour in humans.
As for the other items, the shell is described exacly as you said. Cowry shells are symbols of fertility, birth and wealth, and there are adible species of the genus Leporicyprae, such as the map cowry (Leporicyprae Mappa), both of which would explain the flavor texts on that spellcard. Don't see how to use it for Danmaku though, Kaguya shoots lasers in a star pattern and rings of smaller Danmaku.
The Buddha's Begging Bowl is an actual real-world relic, and in-story it's supposed to give off a shine or gleam "as though set with the most beautiful gems". Not much use as a weapon either, her Danmaku has lasers along a semicircle for the gleam and star Danmaku for the gems ...or something.
The Robe of the Fire Rat is exactly that: A robe made from the skin of fire rats, that would be resistant to fire and heat. In popular culture the only reference I found is that it's also the name of Inuyasha's trademark red clothes that were once owned by his father and share the same qualities, as well as acting as armor and repairing itself. More of a defensive item if anything, but a boss resistant to Danmaku would be a little unfair (except for a Survival Card), and her Danmaku is pretty straigthforward: some laser shots and lots of flames.
The Jewel from the Dragon's Neck is pretty self-explanatory, I couldn't find any further references. The corresponding Danmaku has multicolored bullets and laser shots.
Fun fact: The second Inuyasha movie "The Castle Beyond the Looking Glass" features the items and a character based on Kaguya-hime as well. It also associates the five treasures with the five traditional elements as follows: Hourai Branch - Wood Dragon's Jewel - Metal Fire Rat's Robe - Fire (duh) Swallow's Cowrie Shell - Water Buddha's Stone Bowl - Earth
