If I understand correctly, the main text box on a page is a special box, since in addition to the normal paragraph typesetting, text can also overflow into the next page. Is it possible to, say, have two minipages (or any sort of boxes that I can position arbitrarily in my document) so that the text in the first box that overflows gets put into the second box?
With a lot of manual intervention it's possible, but not with the standard
A "reduced version" of TeX's page builder is available to split vertical boxes:
Now the box register
Careful reading of the TeXbook or TeX by topic is necessary, as this low level mechanism has never been canned into high level macros.