Is copy-editing/typesetting the only step in the publishing process where someone other than the author makes changes to the manuscript?

In general, you are correct: in general the editors and reviewers can only make suggestions for changes while only the author actually makes the changes.

In practice, however, good and thorough reviewers (and occasionally editors too) may often make highly specific suggestions for changes (e.g., "Delete this sentence", "Change phrase X into alternate phrase Y", "Cite papers A, B, and C"). Authors are strongly motivated to accept these changes in exactly or close to their stated form, for two reasons:

1. Doing so is likely to please the reviewer and increase chances of acceptance.
2. The reviewer often has a good suggestion and there's little reason to choose an alternate phrasing.

Finally, note that in some cases copy-editors do significantly more than just formatting/grammar-checks/type-setting. In particular, for publications with very broad audiences, it is sometimes the case that the copyeditor will actually significantly modify the words and phases used by the authors in order to make the text more accessible and more in keeping with the publication's preferred "tone." In this case, though, the authors always have a chance to review the suggested changes and make corrections in case they introduce errors, though objections to style may not be accepted by the journal.

The procedure goes as follows:

1. The authors submit a paper
2. The editor decides whether the paper merits a review (is aligned to the journal topics, is well written, etc.). Then, he selects reviewers and sends the review requests.
3. The reviewers get back with comments. Usually, these comments/corrections/suggestions are mandatory.
4. The authors make these corrections, new tests/results, etc. asked by the reviewers and return the corrected paper.
5. Points 3-4 are repeated until all reviewers are satisfied, or the procedure reaches a maximum revision number (set by the journal usually) at which point the editor makes the call (accept/reject)
6. Upon acceptance, the authors submit the source files of the paper according to the journal rules (latex, docx, etc.).
7. A copy-editor formats the paper and tries to make it look better, checking for typos, grammar, etc. Then, the paper is sent back to the author for one final check. The copy-editor is usually not a specialist in the field, so some times the mathematical formulas could get screwed up.
8. The final paper goes for publication.

These are the steps according to my experience as author and reviewer in the field of electrical engineering.