Is there a rule or other agreement as to how footnotes are spread out (or not) over multiple pages?
Footnotes of, say, one or two lines appear at the bottom (duh) of a page. But larger footnotes, in my observation, appear either at the bottom of one page or at the bottom of two or even more pages. There is no obvious (to me) rule involved; some large footnotes appear on one page – shrinking the real estate for the main text – and some are continued on the next page.
There are various possible practices, but the most common one is for a footnote which is so long that it won’t fit on one page to put part of it on the next page.
That’s just it. “…won’t fit on one page” is open to interpretation (which is the question in the OP). Does “won’t fit” mean the footnote takes the entire page? half a page? all but one line?
I’ve seen footnotes, admittedly large, span two pages which would have fit on one.
“Won’t fit on one page” usually means that you set the main text down to the line including the footnote number or symbol, then see if you have enough room left on that page for the footnote. If you have enough room, put in the footnote, then continue with the main text until you run out of room. If you don’t have enough room for all the footnote, put in as much as you can, and put in the rest one the next page.
I think that’s what MS Word does, though it probably needs to do some small adjustments to avoid widows and orphans.
But there are other methods, some of which try to have the same amount of main text on every page – which means you get a fair bit of white space in some footnote sections. Another method is to try to equalise main text on facing pages.
In general, if the footnote fits on the bottom of the page, it is put there. But there are cases that make that impossible. If the footnote is too large, it may push the footnote number onto another page, which means the footnote comes before the reference. Thus, in general, the footnote goes on to two pages if it would have pushed its reference to the next page.
This is one of the hardest aspects of book design, and one reason why people have gone to endnotes or referencing authors in the text. If you have a page where the footnote reference falls on the very last line on the page, when you make room for the footnote, the reference moves to the next page. It takes a lot of typesetting skill to avoid this.
In general, multipage footnotes are designed so that they fit onto two pages, with the space for footnotes being equal (at least, that’s the goal). Thus if the footnote had 20 lines, you try to put ten lines on each page. It gets more complicated if you have a lot of long footnotes. But the footnote must start on the same page as the reference.
Word uses an algorithm on how to break footnotes, and tries to make the footnote space the same on both pages with a two-page footnote. But this sort of typesetting is an art, and there are no automatic rules to make it work.
In general (there may be exceptions) a footnote will start on the same page as the reference mark. I guess a situation could conceivably arise in which that is impossible as it would put the mark on the very last line. In any case, the rest of the footnote, if it doesn’t all fit on the page with the mark, will be continued onto the next page. I have seen footnotes that go on for pages (I think that is dreadful style, but that’s just my opinion). In that case, you have to choose. You could fill those pages with footnotes (my choice, to be honest) or decree that no page is allowed to contain more than x% of footnotes and fill them with test until that point, and then fill them up as needed.