How do I recognise a peer-reviewed journal article? I have lots of research papers, but can I tell which of them are peer-reviewed. Also, on Google Scholar, is there any way of finding only peer-reviewed articles?
The cynical answer is that in borderline cases it's impossible to be certain what is peer-reviewed without a lot of investigation.  Fraudulent journals claim to be performing peer review, and it's hard to disprove their claims (you have to distinguish between no peer review, incompetent or corrupt peer review, and genuine peer review with a very low requirement for how interesting the papers have to be).  Sometimes someone gets a clearly absurd paper accepted (see SCIgen and follow-up projects such as Mathgen), in which case we know something is terribly wrong.  However, there are other cases where we just don't know for sure, even though nobody really trusts the journal.  It takes a lot of work to investigate, and there are tons of ridiculous "journals" operating on the web, so nobody has the time or energy to look into every case.

On the other hand, this doesn't come up except with pretty weird journals.  Everyone working in a field has a pretty good idea of which organizations and journals are reputable; if an expert doesn't know what to make of a given journal, then that's a bad sign.  I don't know of any systematic way to make this judgment other than based on experience and mentoring, although Beall's list is a useful tool to flag highly questionable publishers.

In particular, I know of no automated tool to do searches only for peer-reviewed articles, and I see no prospects for building a reliable tool of this sort (given that some publishers are simply lying).

This makes the situation sound much worse than it is.  I've never run across a paper I was even remotely interested in where I had any doubt about its peer review status.  However, it's a hard problem if you go dredging through junk journals.

Incidentally, in mathematics it would be a bad idea to limit your reading to peer-reviewed articles.  Preprints are not yet peer reviewed, and if you ignore them until they are formally published, you may never catch up to the state of the art.