Finally got the tables to display according to APA guidelines. It appears that borders only work reliably on table cells, and you can only get them to display different top/bottom from right/left borders by explicitly specifying every style, width and colour for the borders. It was setting the right/left borders to white that finally "suppressed" them. For the bottom border of the table, I had to add a bottom border to cells in the final row.

I then trimmed the space-after attribute for all headings, resulting in text that looked better -- it appears you want gaps between sections, but not between headings and their following paragraphs -- and virtually no instances of widows (which is luck, but the odds are better when there's a smaller space-after).

As far as I can see, I have only one remaining problem: I simply can't find a way to make footnote numbers appear superscripted in the text. According to the FOP compliance table, it should support baseline-shift: super, but it just doesn't work; and vertical-align, which should also be able to take a value of super, fails too. There may be some combination of settings I can use to make it function; I'll need to experiment.