Career options in scientific/academic publishing for PhDs and Postdocs
I'm in a similar situation, meaning that I'm also a recent (less than two months) Ph.D. graduate and currently focus my effort on a job search (both in industry and academia). The differences between you and me include the discipline (not essential) and my lack of postdoctoral (or any other, for that matter) job offers at the moment. Since I briefly have been considering a temporary career move toward a scientific publishing world, while I do not have a direct work experience in the scientific publishing industry, I think that I have browsed through enough job advertisements to form and share my opinion on the subject, as follows (certainly, take it with a grain of salt).
I consider the temporary career move toward scientific publishing that I've mentioned mostly as an industry career option and not an academic one. While in the context of scientific publishing, there is an intersection between classifying such career option as one of industry or academia, I believe that a career in scientific publishing cannot be considered as an academic career. Therefore, my first point is that, if you, as myself, consider academia as your main ultimate career avenue, going the scientific publishing route is not a good option (unless it is very short-term and due to circumstances). Continuing my road terminology analogy, the above-mentioned avenue and route are not parallel, thus, after some time traveling the scientific publishing career route, one can find themselves too far from the academia career avenue. That will most likely make returning back to academia very difficult, if not impossible.
Based on various job descriptions that I've looked at, a career in scientific publishing seems to involve much less scholarly activities at the expense of much more project management, marketing and operational ones (especially, for non-senior editorial roles). Moreover, internal organizational politics, tight deadlines and other time-related issues, as well as stress of juggling multiple publication projects add insult to injury.
Finally, I personally dislike the scientific publishing industry's negative role due to lack of openness in dissemination of scientific information and corresponding strategies and tactics they use (that includes their "implementation" of the open access paradigm). Therefore, it would make even more difficult for me to work in such environment and organizational culture.
Considering all the above, I would not go the scientific publishing route, even temporarily, but, instead, would try my best to join academia as a postdoc or junior faculty member and progress there or, alternatively, go the real science-related industry route (less perfect, considering the ultimate goal, but, can be either temporary, or long-term, combined - in a form of consulting - with academic positions).
I used to work in academic publishing. In the following answer, job titles are very variable; the word "editor" can mean one of a multitude of jobs.
Q1. What kind of full-time career paths can be envisaged in the publishing industry?
At the bottom tier there's the proofreader. You read the paper and make sure everything is in the right format, e.g. perhaps your journal follows a style where the word 'datum' is used as the singular form of 'data', and it's your job to fix it if you spot it. You don't really do copyediting, you only do proofreading. To do this job you don't really need a PhD, or even a Bachelor's. It's also (in my opinion) very boring.
Next up there's the journal production editor. When the editorial board accepts a paper, these people get it through to published form. This could involve copyediting, typesetting, sending to author for corrections, uploading online, etc., or (rather more likely) working with the people who actually do these things. The JPE acts as the overall coordinator and is expected to be able to answer author questions "my paper was accepted three weeks ago and I've not yet received proofs, what's going on?" For this job you'll typically need a Bachelor's degree.
Then there are the desk editors. They can do all the things JPEs do, and then some: they might for example be responsible for the content on the journal's website, they have to work directly with the editorial board (e.g. editorial board says let's take advantage of [this conference] to promote our journal, the DE is then responsible for getting management to approve the expenditure, work with the artists to put together flyers, posters, etc.), handle author questions "my paper was submitted three months ago and I haven't heard anything, what's the review status?", etc. There's also a good chance that desk editors handle books (JPEs usually don't, because they can easily handle papers from many different journals at the same time). Desk editors usually need a Bachelor's degree, but it is also a somewhat more advanced position than JPEs.
Next up is acquisition editors, who are tasked with acquiring new projects for the publisher. This could be new books, new authors, new journals. AEs write to academics and convince them to publish with [publisher]. They could conceivably be asked to attend conferences to promote the publisher's products. The better desk editors can conceivably also do the AE's job, but this is usually a more senior position for which an advanced degree is preferable - a Masters or PhD. Because of this advanced degree they can also be asked to resolve peer review disputes, make a final decision on a publication proposal, and so on.
This covers only the editorial jobs. Of course at various levels you also need marketing executives (who market the publisher's products), HR, people management, and so on, but these other positions are less specialized to the publisher and you'll have them in other industries.
Q2. How is the career progression? What kind of challenges does one typically face in this profession?
Progression kind of stops with acquisition editors - once you become an acquisition editor, the "next step" is doing it more, in more fields, manage other people doing it, and so on.
Right now I'd say the main challenge is dealing with people who think publishing is a scam and that you, by working in it, are complicit in the scam.
Description of typical day. Based on the description this person is at the interface of desk editor and acquisition editor - she both acquires projects and produces them.
Q3. Are there any short-term jobs or internships suitable for PhDs/Postdocs in this field?
You'd have to ask, but in my experience, no. You could get short-term jobs or interns as a Bachelors student, but not for the more advanced roles performed by PhDs and Postdocs. It's not something which you can easily hand off to an intern and then pick it back up when the internship is over.
Q4. What are the typical exit options? For instance, is it easy to switch back to your research career (postdoc/professorship)? What kind of secondary and tertiary industries, if any, be interested in hiring someone with an experience in the world of publishing?
Like with any field change you'll need to convince someone you have transferable skills. If you remember your research skills you could move back, although of course the longer you wait the less you'll remember and the harder it gets. Moving to another industry? Same thing: you could argue you can work under tight deadlines (if you did production), you are good at networking (if you did acquisitions), you can handle multiple different projects simultaneously, etc.