Does the word for “flesh” that reappears here also reappear in the Hebrew? What does it mean to say husbands and wives are “one flesh”?

Yes, the word is בָּשָׂ֖ר, basar. Obviously it means that husbands and wives are to be so close as to be a single organism with utterly united interests. The full meaning is not given here but is explored throughout the Bible. The origin of the metaphor, however, comes from this story, that Eve was made from the flesh of Adam.