Sagume Kishin: She’s a high-ranking Lunarian and a god. Both ULiL and the invasion in LoLK were planned by her. Essentially it’s an emergency evacuation of the moon because Junko (the final boss) is laying siege the Lunar Capital with impurity. The plan was to recreate the Lunar Capital in Gensokyo as a last resort. In the meantime, the Lunarians are living in the Dream World (but most of them don’t know that because they recreated the Lunar Capital out of dreams).
The connection to ULiL comes from her power. She’s basically a super amanojaku whose words are always lies. Not because she’s a liar though, but because the world hates her and will change itself to make what she says false. If she says the sky is blue, it will turn red. This is a terrifying power, so she speaks very carefully and never makes public appearances. Anyway, she put a fraction of her power into the Power Stones of ULiL, which allows them to make false words (urban legends) into reality. The ultimate plan was to spread a rumor about the Lunar Capital, which would materialize the Lunar Capital in Gensokyo and kill everyone who lived there. But she wasn’t planning on actually carrying that out. It’s just insurance. When the protagonists show up with Eirin’s drug, she recognizes them as part of Eirin’s plan and lets them pass.
Clownpiece: She’s a fairy from Hell who was sent to the moon as part of Junko’s plan. Technically she’s a minion of Hecatia (extra boss), but they’re all working together. The moon has no life or death, so it has no fairies. Conversely though, bringing fairies to the moon will bring life and death to it. Junko plans to lay siege to the Lunar Capital by threatening it with lifeforce. Clownpiece has been “purified” by Junko’s power and is now pure lifeforce. She’s basically just hanging out on the moon being a fairy, for the sole purpose of pissing off the Lunarians. She lives outside the Lunar Capital, so there’s no air or water or anything, but she thinks of it as a relaxing vacation from Hell.
Edit: It’s been pointed out to me that Clownpiece’s power (”to drive people to madness”) is very similar to the
lampades, which are nymphs under the command of Hecate. So she’s most likely a Greek nymph, which is apparently the same thing as a fairy.
Junko: A former Lunarian turned embodiment of vengeance through her own power: to purify anything. She had a grudge against Chang’e (Lunarian who doesn’t show up in this game, eternally imprisoned in the Lunar Capital) because of the death of her son and her husband, but then purified her grudge until the reason didn’t matter anymore. She can’t get to Chang’e as long as she’s being held in the Lunar Capital, so she came up with a plan to force the Lunarians to move, giving her an opportunity to attack while Chang’e is in transit. She enlisted Hecatia’s help for this plan. She’s kind of a strategic genius or something, who was bored by how predictable the Lunarians were until the protagonists showed up.
Hecatia Lapislazuli: Goddess of Hell, presumably the Greek Hecate. Necessary to Junko’s plans because she has unlimited fairy minions and can freely travel to the Dream World and harass the Lunarians there with more lifeforce (since Junko predicted they’d run there). She can freely travel to the Dream World because she has three bodies in three worlds at the same time (the Moon, the Earth, and the Spirit/Dream World). She agreed to help Junko out of sympathy for what she lost, and also because Chang’e’s husband shot down the sun way back when, which hurts Hell goddesses because you need light to have shadow. Also she’s related to Apollo, who is also the sun, so there’s some weird multicultural crossover myth vengeance going on.