Wow, not only was this a sign-heavy episode, but also a note-heavy one! I’m just going to go through them in no particular order.
Yukichi is the guy on the 10,000 yen bill. If it were in America, the joke would be Benjamins.
Oshikura manju is this goofy Japanese kids “game” where they link arms in a circle back-to-back and push in and out. The movement is supposed to heat you up. See also: Maho Girls
Kanpu masatsu is that thing where you rub a dry towel quickly over your body to heat it up. You can probably visualize a wrinkly naked old man doing it if you’ve seen enough anime.
Hottokeya is a pun (or not) that sort of combines the word “hot” with the verb “to leave alone”, so it’s as if the person in the hot spring is telling others to let them be.
The flag on the moon is a total parody of Potepo from Aikatsu. Don’t try to convince me otherwise.