As a reviewer, what steps can I take after paper is published despite major scientific and ethical concerns, with no explanation from editor?
There may not be much you can do about the scientific issues.  A profoundly wrong paper should be retracted, and a correction should be issued for anything serious that's objectively wrong and correctable.  However, the problems you've identified (poor methodology and interpretation) may not definitively fall into these categories, depending on what exactly you mean.  There's no ethical reason why a journal can't have low standards or publish controversial papers.  The primary risk is that their reputation will suffer if they publish foolish papers.

It would indeed by unethical if they claim to be a peer-reviewed journal but aren't taking the review process seriously at all, and that sounds like what's going on here.  However, it would be difficult to prove based on an outsider's view of one case.  I imagine that if pressed, the editor would claim he/she evaluated the paper and your review and decided the paper was more credible.  It would be hard to prove that there was an ethical problem, rather than just poor scientific judgment on the part of the editor.

On the other hand, the allegations of copyright infringement and self-plagiarism are ethical issues that transcend journal standards or policies.  That might be a good place to begin, and perhaps more will come out during the investigation.

Here are some thoughts:


Your proposal to share this information with the editorial board of the previous journal sounds like a good idea.  I'd contact the publisher directly as well as the editorial board.
There may be an ethics committee in a relevant professional or learned society.  They could conduct an impartial investigation, which might lead to public censure or a retraction.
You could report these papers to the authors' funding agency, if any.
You could report these papers to the authors' employers.
You could publicize your experience on the web.


Among these possibilities, #1 and #2 seem most likely to make a difference regarding the journal, #3 and #4 might lead somewhere with the authors but wouldn't address the problematic nature of the journal, and #5 could be a valuable public service but is unlikely to change much by itself.