That routine on page 10 is what old style gangsters would do when introducing themselves, it was extremely ritualistic. They were where yakuza came from, back in the 1600s they were called tekiya and bakuto. The kids here doubtlessly picked it up from period TV shows (from the very few tvs in town) or traveling plays, since they didn't really have access to movies.
That 50 yen coin would be worth about 350 with inflation, about US $2.50.
So I'm guessing from the ending pages that the other kid was so upset that it didn't work that he's run away from home and Akihiro has to deal with everyone wondering where he is without revealing their working (he thinks) time machine?
Thanks for the chapters, they're always a treat!