Keito uses the word "ichimaime" to mean the "leading role." I left it as "ichimaime" because the word is said to originate from Kabuki theater, and I suppose it's a very Akatsuki thing to say.
Scroll of Wind - 3
A "yuru-character" or ("yuru-chara," etc.) is a type of mascot character that usually promotes some kind of business, geographic area, or industry. Their designs are usually simple and cute. The yuru-character that Midori refers to as "Ku*mon" is "Kumamon," a famous mascot of the Kumamoto prefecture.
Hattori Hanzou was a famous ninja in the Sengoku era. Shinobu also mentions him in Christmas Live.
Scroll of Clouds - 1
"The Tale of Jiraiya" Jiraya, Tsunade and Orochimaru are legendary characters with the power to control toads, slugs, and snakes, respectively. If you've watched Naruto, you get the idea. According to the tales, snake magic is strong against toad magic. Toad magic is strong against slug magic. And slug magic is strong against snake magic. That's the three-way stand-off that Shinobu is talking about.
"Hiko*yan" is "Hikonyan", a yuru-character mascot for Hikone, Japan.
"Akazonae" is a type of red lacquered samurai armor that was typically reserved for high ranking or valiant fighters.
Signs of the Coming Storm - 1
Kuro occasionally uses "danna" to refer to Keito. "Danna" is a word for the master of a house in the old sense of a house, including the family members and servants, etc. It can also be used to refer to a primary patron (as in the patron of a geisha) or a person's husband/actual head of household, but here Kuro is referring to Keito as basically the patriarch of Akatsuki, I believe.
Souma talks about "juuji-kappuku (十字割腹)" in this chapter... I think he is referring to "crosswise seppuku" in which the person cuts their abdomen both horizontally and vertically, although I can't find the exact word he uses on the internet..
Keito talks about "challenging dojos." That's dojo yaburi , a tradition of barging into someone else's dojo and challenging the persons within.
Souma cautions Keito that they are "within the castle." That's "denchuu de gozaru(殿中でござる)," referring to the inner citadel of Edo castle, and in saying it, he is hinting at a famous rule that one may not draw their sword within the inner citadel. Basically, behave properly.
Signs of the Coming Storm - 2
Keito thinks Chiaki is talking about the koi fish and not the type of love called "koi"......
A Hero in Stormy Times - 1
"Nichibu " is a broad genre of traditional Japanese dances.
A Hero in Stormy Times - 2
"Katon no Jutsu" (roughly "Fire Release Technique") and "Suiton no Jutsu" (roughly "Water Release Technique") are common names for ninjutsu related to their corresponding elements, but they don't have any particularly well-defined meaning in terms of what they actually do.
Shinobu is referring to a saying about ninjas, in which having a sword over their heart indicates their resolve to endure.
Epilogue - 1
Souma talks about "manjuu kowai" in the end of the story. That is a common rakugo (a type of Japanese comedy) story about a man who claimed to be afraid of manjuu (sweet bean buns) in order to trick his companions into presenting them to him to try to frighten him.
Souma's final words are お後はよろしいようで ("oato ha yoroshii you de"), which is what a rakugo performer says when he is getting ready to leave the stage because the next performer is ready to go up.