00:00 The Lover of Princess Kaguya

This musical is based off the manga side-story of the same name, which the Sailor Moon S movie is also based off.

13:40 Marital quarrels are raw qilin meat

"Kirin" can refer to "qilin" or "unicorn". This line comes from the manga, it's a nonsense non-sequitur.

19:11 Ar

Ar is short Artemis, if that wasn't clear.

21:19 Princess Kaguya

The Tale of Princess Kaguya (the Tale of the Bamboo Cutter) is a Japanese folktale involving an old man who finds a princess from the Moon in a bamboo stalk, her beauty attracts five suitors and attention from the Emperor, though invariably at the end of the story, Princess Kaguya returns to the Moon.

21:19 Kakeru Oozora, Himeko Nayotake

These names have pretty clear meanings behind them. 
Kakeru = Soaring
Oozora = Heavens
Himeko = Princess-Girl (as in Princess Kaguya)
Nayotake = Bamboo

29:11 Kuro, Mike, Tama

These are all common cat names. Kuro means "black", mike (pronounced "me-kay") refers to tortoise-shell patterns and usually is a name given
to calico cats. Tama means "ball".

34:34 Yuzu

A tasty citrus fruit mainly cultivated in East Asia. It is a common ingredien in Japanese cuisine, its zest and juice often beng used in the same way lemons are used in other cuisines.

36:50 Sugar stars

This is konpeito (from the Portugese confeito), a popular sugar candy in Japan. They come in multiple colours, but they usually are not flavoured. They are referred to as "sugar stars" in the translation to make the connection between the konpeito and the space theme of the musical closer.