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 Yummy" 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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