I don't understand this response? Kaguya took the elixir, so she was executed for it, that is her punishment. However, that didn't work, because of her power, so they sent her to Earth. Here is her profile, http://en.touhouwiki.net/wiki/Kaguya_Houraisan. This is mentioned there.
And Kaguya is a what? A human? No. She's a Lunarian. The law forbidding the ingestion of Hourai Elixir applied to her too, princess or not. The question was : "How would the Lunarians react if Mokou ever went to the moon?" Mokou is a human. Lunarians used elixir to tempt the humans from Earth, they never mention that humans are forbidden to drink it. Although they probably never did, because they were deceived or the task necessary to obtain it was like Kaguya's impossible request. The question is a "what if" scenario, but this scenario is not unprecedented. The Moon has already had visitors in the past. Mizue no Uranoshimako - kept alive and under false information that he was in the Dragon's Palace, then sent to Earth, hundreds of years later and killed off by Eirin's little gift. Then there was the 1st and 2nd Genso-lunar war. In both cases, the intruders were beaten and sent back (casualties unknown). But they kept Reimu up there in order to clean the suspicion off their name. Then sent her back to Earth. So if Mokou somehow ended up on the Moon, they could: a) send her back b) try to kill her, fail, and then send her back or c) lock her up
What follows is just my theory, so don't quote me on it: Eirin claims that those who drink the elixir take on the same taint as the humans. (CiLR chapter 1). If this is the case, having an impure being on the Moon for a prolonged period of time could result in tainting the Moon's perfectly pure environment and thus, threatening the infinite lifespans of all its inhabitants. This is contradicted by the Lunarians' desire to bring both Kaguya and Eirin back to the Moon. But perhaps it is due to their desire to have the Sage of the Moon back home as well as the princess, who was believed to be punsihed enough. But there may be another reason why they wanted to do this. They didn't want to allow to create any more Hourai Immortals. After they saw how Kaguya was being a little more generous about giving people like the emperor or the old bamboo cutter a dose of the elixir, they decided it would be better to isolate the princess from the humans and bring her back, despite being as impure as them. But even if Kaguya doesn't share the elixir with anyone anyomre and even if Eirin refuses to make any more, there is still a way for every single Earthling to gain immortality. As unlikely as it sounds, by eating the liver of a Hourai Immortal. This creates a possiblity for another potential invasion, led and fought by immortals and if I was a Lunarian living in the capital, that would unsettle me, to say the least.
1) Has Reimu ever really done any real extermination to any youkai? If not, why do you think is that so.
As far as I can recall she only beat them up and just leave them be (although I kind of admit they don't quite pose as a real threat to Gensokyo ot its inhabitant mainly human to a certain limit that it). The one I can see Reimu really goes for extermination are the one in Touhou Suzunaan ~ Forbidden Scrollery.
Her usual "beat the youkai up with danmaku" is what she refers to as extermination, which is always just a temporary matter. She doesn't ever permanently exterminate any youkai. And fairies don't count, as they are naturally immortal anyway.
Reimu does not really exterminate youkais, because she has nothing against them. You can say she has no innate bias against youkai, even if it is her duty to exterminate them.
No bias?
"From now on, I'm going to exterminate every youkai just because they are youkai!" - Reimu Hakurei, WaHH ch. 8
"I'll beat anyone who's a friend to youkai!" - Reimu Hakurei, UFO, stage 6
Kyouko: "It's a temple for youkai that is kind to youkai"
Reimu: "Well, then I'll have to exterminate you right away!" - Ten Desires, stage 2
"I was in the middle of indiscriminately exterminating any youkai I find." - Reimu Hakurei, Phantasmagoria of Flower View, stage 7
""On the way back, a youkai was just sitting there minding its own business. And it was enjoying a book, too!... I tried to exterminate it with a surprise attack, but it retaliated! It was pretty cocky and strong. I let my guard down, so I didn't expect shots to come from behind..." - Reimu Hakurei, Curiosities of Lotus Asia, chapter 1
She treats everyone the same. However, since her job is youkai-hunting, she acts particularly strict with youkai, but she doesn't really care much about humans or youkai. This is a recurring line in Reimu's profiles ranging from MoF up to TD.
