ALL EPISODES
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Lersant - Shikanoko often uses the made-up word "Notsu" (のつ) as a greeting/farewell and the like. This is the Japanese word "Tsuno" (つの/角) with its two syllables flipped around. It originally means "horn-like appendage", and thus applies to the antlers of, you guessed it, deer. So, Tsuno (JP) becomes Notsu, Antlers (EN) becomes Lersant. She also mixes the word into some familiar phrases in a punny fashion, which we have translated accordingly (i.e. "Notsukaresshita" , a spin on "Otsukaresshita" (おつかれっした, or "Take care", becomes "Take caribou").
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EPISODE 03
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The names Koshi-tan gave to the potted plants are historic references. Nakahama Manjiro, AKA "John Majiro", was a key translator during the opening of Japan. Iruka, Umako, and Emishi are all part of the well-renowned Soga clan. Soga no Umako is Soga no Emishi's father, and likewise, Soga no Iruka is Emishi's son.
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EPISODE 06
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The "Yummy, Dummy" segment is a translation nightmare. "Uma" can mean both "Yummy" as well as "Horse". (Bashame's name itself refers to a horse-drawn carriage/cart.) "Shika" of course means "Deer". So, the two are enjoying their meals while making their own animal-themed utterances of enjoyment. This is why Crunchyroll went with "Ta-hay-sty" and "Deerlicious", which is a valid enough approach itself. However, on top of this, the true punchline is that the kanji for "Horse" (馬) and "Deer" (鹿), when placed next to each other, form the all-too-familiar word "Baka" (馬鹿). That is to say, "Idiot". The scene is, in the end, about how the two dummies enjoy their yummy meals.

Wink Murder is the most common English term for a party game based around, well, killing people via winks (The Japanese name is Wink Killer). The exact rules differ between groups and cultures, but the punchline of the title of the segment that refers to it is that the game uses winks as the form of execution to make it easy to be covert and thus have a mystery to solve, but her antlers (audibly) bending would give her away instantly.

The Tetsuwan Dash (Often stylized as THE TETSUWAN DASH) is a long-running variety show about the members of TOKIO, a band, tackling all sorts of challenges relating to several facets of life, including sustainable living via farming. At least one of the episodes of the show has covered rice blight, as mentioned by this anime episode. Here is a writeup from their website on the subject, as well:
https://www.ntv.co.jp/dash/articles/65k20pimq1u2y06zri.html
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EPISODE 08
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The title of this episode, "ゆくシカ、くるシカ" (One Deer Comes, Another Goes By) is a reference to the long-running New Year's Eve NHK program "ゆく年くる年" (One Year Comes, Another Goes By). As the name implies, it is about looking ahead at the year to come, while also reviewing the major happenings of the year that's ending and focuses on the New Year's Eve festivities currently occurring.

Omikuji is a traditional form of fortune reading, of which Great Luck is the best outcome one can get. On the other hand, Luck to Come is a so-so result. When Koshi-tan is unfolding her omikuji slip, it initially looks like it will say "Great Luck", hence her reaction, but the fortune reveals itself to be "Great Antlers" instead.

~05:57 - The "I need to tie away this misfortune, fast!" and "We can go tie it up. Don't worry." lines are a reference to the belief that tying up one's Omikuji will leave the effect at the shrine, so the bad luck is stuck there rather than to the individual.

~20:53 - The newspaper signs are as follows:
(top left) [The Weekly Deer]
(top right) [The Weekly Deer] 
(big white text) [Popular Deer Idol Faked It To Make It?!]
(white text on circular badge) [The Shocking Scoop]
(top left under photo) [There are also rumors that Miss Koshi used to be a delinquent.]
(LINE chat parody)
[Koshi Torako (Capybara)]
[Today]
[Can you get me 10 sets of stick-on antlers by the weekend?]
[That'd be pretty tough, this week's had a lot of orders.]
[I can get you 5 by the weekend, 5 more by next Wednesday.]
[That's fine. Thanks in advance!]
[They always make sure no one knows I'm not a deer, and are so comfortable and easy to wear, they're such a lifesaver!]
[All of our products are 100% pure material with no compromises, after all.]
(black text, underneath) [Leaked SHIKINE Screenshot]
(right side, undrneath photo) [Miss Koshi Torako, Antler-less (17)]
(right side, underneath rightmost photo) [The miraculous photo that lead to her becoming a hit sensation] 
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