How often do publishers change/tweak articles to achieve visually appealing results? When reading this question it seems the theoretical answer is that LaTeX is not perfect and manual/human intervention is required. In the TeX Book Knuth says:


  But the problem of page make-up is considerably more difficult than
  the problem of line breaking that we considered in the previous
  chapter, because pages often have much less flexibility than lines do.
  If the vertical glue on a page has little or no ability to stretch or
  to shrink, TeX usually has no choice about where to start a new
  page; conversely, if there is too much variability in the glue, the
  result will look bad because different pages will be too irregular.
  Therefore if you are fussy about the appearance of pages, you can
  expect to do some rewriting of the manuscript until you achieve an
  appropriate balance, or you might need to fiddle with the
  looseness as described in Chapter 14; no automated system will be
  able to do this as well as you.


In the LaTeX Companion Mittelbach writes:


  For this and other reasons, getting the final layout of the book was fairly labor intensive and even required minor rewriting (on maybe 10% of the pages) in order to avoid bad line breaks or page breaks.


Ignoring the fact that LaTeX is not the only software that publishers use, I am curious if the theoretical answer is actually the practical answer. Specifically, I am interested in:


How often do journals make manual tweaks to their templates/layouts
when typesetting articles?
How often wording of the text is changed to increase the visual
    appeal of the article?
Since some journals provided me not only the final typesetted result but also additional documents that captured the editing (for example a scanned document with the remarks or the copy editor or even a diff of the "source code" (actually a mix of text and source code for the formula)) I can say, based on this sample size of about a dozen: Always.

To be more precise: The text has been changed always, but I can't really tell if this was for better visual appearance or because of the text. Also the figures have been shifted  around in all cases (and I guess that this was for the visual appearance). I can't say whether the journal tweaked their template (because I don't care too much about the templates in the first place).

Maybe I should add that I have been more or less always pleased with the editing. In most cases I did not notice the changes from the typesetted version, but only after a closer look at the diff. At a few places the editing introduce a wrong wording (most cases there was a note to check this passage) and in some cases the typesetting of the formula was worse.