はいります This one I have translated as 'entrance.' The picture doesn't help me here.
Looks to me that the picture is of a geisha (?) welcoming two guests coming into a place/business. So, I think "Entrance" is probably accurate.
I'd say "Golden Boy" and "Peach Boy" are fine, but "Urashima" alone seems to be kinda missing the boat, as it were. Especially because its's about the man and not the place. Unfortunately, space is at a premium, so "Taro of the Inland Sea" isn't gonna work. At an absolute worst-case scenario, I still think something like "US Taro" would be a better name, though. Still not good...
This is in the upper right corner of the Quick Picross screen. I have no idea what 'wanko' is supposed to mean.
Cats meow and dogs bark in the west, and in Japan they nyan and wan, respectively. Nyanko and Wanko are cute names used to refer to a cat or a dog. Li'l Barky or Li'l Woofy, basically. Simply Dog would be fine, but not so much fun.
This is a play on words from "The Seven Samurai," but instead is "The Seven Samugariya." "Samugariya" is a person who feels cold easily, the type who have a blanket on their lap right up until summer starts.
"Seven Freezing Friends" or something?
God, this stuff is obscure. Comes from the idiom "whispering prayers to a horse," to mean that it's a waste of time to bother to explain. See 馬の耳に念仏. But instead of prayer (nenbutsu), it's clay (nendo).
So, "putting clay on a horse's ear" or "clog up a horse's ear"
Finally one easier to understand! The Japanese title for "Alice in Wonderland," but reads "Alibaba in Wonderland" instead.
From the famous book by Soseki Natusme, "I am a Cat." But instead of cat (neko), it uses tumbling (nekorondeiru). So, "I am tumbling"
Looks like a play on the Japanese title of the song El Cóndor Pasa. In Japanese it reads "A condor flies" (コンドルは飛んでいく), but replaces "flies" (tobu) with "not a black kite" (tobi ja nai). So, "A condor isn't a black kite" (a black kite is a time of bird).
Sorry, dunno what the plan on words is here. Just means "boy's are quite a thing!"
Looks to be a popular phrase, but dunno the history of it.
"Going Dutch with Mario," meaning to split the bill on dinner. Play on words, because to "go dutch" is called warikan, which sounds like wario.
From the well-known Japanese phrase-turn-manga-turn-anime-turn-drama. "Hana yori dango" (boys/dumplings over flowers), meaning to want something practical over something simple like flowers. But instead of dango (dumpling, but "boys" in the manga title), dancing queen.
"Dancing queens over flowers."
From the phrase Hate a priest, and you will hate his very surplice.
, but instead of "...you will hate his very...," it's been replaced with a judo move, kesa-gatame. So, basically:
"Hate a priest and he'll take you down with a kesa-gatame"
Gonna leave the rest of these for now, but I think you can see that it's not really useful translating this, since they're mostly puns. From what I read in Japanese, it's the number of letters they're counting in the Japanese game, so you may just want to come up with your own random puns. This is the big difference between "translation" vs. "localization." I think rewriting would be more fun for English speakers here.