"Tonari no ie ni hei ga dekitatte ne." (Say, they've built a wall (hei) next door.)
"He~i." (the Japanese 'oh hey what about that, really?' expression).

Yes. That's it. That's all of it. That's the joke, see: hei and heiiii.

"Gingin ni notteru kai?" (Is everybody happy?)
"Ie~!" (Either 'no' or 'yay')

'Gingin ni notteru kai?' is some old (meaning 70 or 80's maybe) comedian's way of getting a response from his audience, the exact equivalent apparently of 'Is everybody happy?' Evidently gingin alone has no meaning, or if it does my informant doesn't know it. Together it just means to be in bursting good spirits, and the 'kai' makes it a question. 'Is everybody happy' gets the answer 'ie--' which can be a protracted no but which can also be 'yay!!!'--the acclamation of the burstingly good spirited audience.

Atarimae da no kurakka (Well naturally cracker)

Atarimae da no kurakka, from what I understand of my informant's explanation, was one of the buzzwords of that 'long ago' period--again, 80's probably--and consisted of throwing totally unrelated terms together. Here it's 'cracker' added on to the phrase 'Well, naturally'. Evidently people also said 'chonmage sumimasen' a lot too. Chonmage are the scalplocks you see employed samurais
wearing in jidaigeki. Scalp shaved, and the long back hair is tied back and oiled so it forms a thick sausage which is brought forward so it lies on top of the scalp.

Panthea: "This is reminiscent of word association football. (Monty Python fans will understand--all others just ignore.)"

Superu ga muchakucha da yo (My spelling's a mess.)
Yurushite kudaseii o-deikan-sama (Please forgive me, o-daikan-sama.)

These are comments from Fuji herself. The second is a phrase from the
jidaigekis (historical dramas) in proper rough male jidaigeki-speak. Properly
'yurushite kudasai, o-daikan-sama.' A daikan was some official in the Tokugawa bureaucracy, a country magistrate. (In the jidaigeki they're always corrupt and venal and grinding the faces of the peasants.) I assume there's some awful pun embedded here too, but my informant had no idea what.

Wake wakame do yo (Seaweed makes no sense [we're just making this up by now])

Possibly a pun on wake wakanee--malespeak for wake wakaranai--'makes no sense.' Here substituting wakame, seaweed. This put my informant in mind of a popular song from long ago, consisting of food names in slightly ridiculous forms,
which as it happens a couple of little Papuwa beasts sing as well, which is so natsukashii no-one can stand it. The 'do yo' is a puzzler. Emiko's suggestion is that Fuji got her letters wrong and meant da yo. Or it could be a pun....

"Abayo" (Sayonara)
"Kebai." ('Kay, bye.)

Abayo is a rough male way of saying sayonara, also very jidaigeki. To which someone else responds 'kebai' which looks like it should be a Japanese word but isn't. It's "'kay, bye."

Umeboshi ga ume (Dried plums are delicious!)

Ume is colloquial for umai, delicious. The pun, of course, is in the similarity of the words. Which sheds sudden unwanted light on those flies in Papuwa who go around saying 'Hai ga haii!' (The flies are high) not to mention the horse that snorts 'uma ga umai' (the horse is clever- another meaning of umai).

Runrun kibun (High spirits)

Runrun kibun means 'high spirits', but may be a pun on the English run. (Run from DOS? C:\DOS run. Run DOS run.)

Then I believe it's 'Maybe I should stop here. There may be people around, who knows?' It's a little odd, so I'm not ruling out yet another pun. Ends with gokurousama, 'thanks for all the hard work.'