How often are articles submitted to peer reviewed journals, not just with made-up references, but with *ridiculously* made-up references -- for instance copying verbatim a how-to-write-a-reference example from the Chicago Manual of Style?

I've never seen it, and you wouldn't be able to get away with it.

I've never seen it. And I've seen a lot of journal submissions.

The best example is the http://en.wikipedia.org/wiki/Sokal_affair.  While it did not involve a peer-reviewed journal, the victim was intensely serious. The Wikipedia page on Sokal links to numerous other hoaxes perpetrated on peer-reviewed journals. Many errors make their way into peer-reviewed journals.  This happens most often when the original source makes a bibliographic error, for example citing to an incorrect page number.  These are sloppy but harmless. What happens next is not so innocent.  Later writers cut and past bibliographic entries as their own, obviously not having checked the sites or read the materials.  This is at best sloppy research and at worst plagiarism.  The best peer reviewers know about these errors in original sources and identify later articles as poor or dishonest research. Some writers and publishers "salt" their publications with a few inconsequential errors so that they can detect later plagiarism.  This happens frequently with other types of material.  Mapmakers add tiny nonexistent entries to detect those who copy their maps.  Lists of contacts sold for advertising contain false names that come back to the owner, thereby showing that the list rented for a single use has been used again. In fact, while I am familiar with many of the links in the Sokal Wikipedia article, I did not read them and therefore committed at best sloppy research.  I have, however, read the Sokal article, which is full of nonsensical material that any serious reader would detect in an instant.

Making up a reference to support an assertion in a paper would be serious scientific misconduct, unless it was very obviously flagged as a joke (science seems to have lost its sense of humour in recent years so you cannot assume that somebody will know you are joking).   I have never seen a fake reference as far as I know, and I'm not sure what the motivation would be for putting in something silly.  However as a peer reviewer I don't usually check references unless I'm dubious of a claim, perhaps I should tighten up on that.

I have seen it only once, and that was in a very old chemistry article (as in 1900-ish). Even then, I think most of the papers were real - the author simply couldn't be bothered to look up the actual page numbers and such, so made up something plausible instead. In the digital age, you'd be found out immediately if you tried this.