British tabloids are sort of a weird thing, even compared with the kind of rags I see in American grocery stores all the time. Yet I see British coworkers of mine refer to these tabloids again and again, either as a source of scorn or with some reverence (as in, they're considered authoritative on some things).
Where did tabloids come from? Do they have any relation with the US tabloid industry? Is there some law that allowed them to proliferate and become so profitable?
Do they have any relation with the US tabloid industry?
I can speak to this part of your question, in two parts.
\1. The American press might have more of a "tabloid"* history than you think. Some of Edgar Allan Poe's first publications included far-fetched stories in New York Sun - the most famous being published nowadays as "The Balloon Hoax". In 1844, it was printed as a sensational news article under a headline that started with the words "ASTOUNDING NEWS! BY EXPRESS VIA NORFOLK: THE ATLANTIC CROSSED IN THREE DAYS!", and sold lots and lots of papers.
The idea of newspapers being sacrosanct bearers of truth, with their own elevated code of ethics, might be more of an early 20th-century phenomenon than it seems. I quite like what Josh Marshall has written on the way the economics of local papers' mini-monopolies really made balanced coverage and unimpeachable ethics a necessity. (He's a journalist trained as a historian, so takes a long view.)
\2. Specific to the American tabloid industry as it is today, there was a very conscious effort to turn it into a reflection of the British industry. Through most of the 20th century, North American tabloids were produced by two publishers: Mike Rosenbloom's Globe group (based in Montreal, then in Boca Raton, Florida) and Generoso Pope's National Enquirer family of papers (started in New York, flourished in Lantana, Florida).
Starting in the 1970s, Pope made a conscious project of hiring as many hard-nosed British (and British-colonial) reporters as he could, sometimes paying exorbitant salaries to recruit them. It was also quite common, in the days before the Internet, for reporters in Florida to sell stories back to the British tabs on a freelance basis - the London Sunday Mirror wasn't considered competition for the National Examiner, so there wasn't really much of a problem when an articles editor in the States rang up an old buddy back at the overseas desk on Fleet Street with a hot story, as long as they didn't publish first.
You can read more on this on an academic level in Tabloid Valley: Supermarket News and American Culture by Paula E. Morton and The Godfather of Tabloid: Generoso Pope Jr. and the National Enquirer by Jack Vitek. There are also dozens of ex-reporters' memoirs out there - I know Jim Hogshire wrote a fairly sensational one. In fact, I mentioned him and Vitek and Morton when I wrote this answer about Elvis sightings a couple of years ago.
I hope someone else can chime in here with more on the "Where did tabloids come from?" part of the question. There's some interesting stuff there - some of the first tabloids were marketed primarily to women and blue-collar laborers - but I'm not sitting next to my bookshelf, and half of the media-related stuff I had on that back in the day has wound up being given away anyway.
* (I put it in quotes because technically, tabloid is a paper size... which was named after a pharmaceutical innovation, which in turn was named for Moses' original stone galleys of the Ten Commandments. Henry Wellcome made the first "tabloid" medications using a bullet mold, but wanted customers to associate them with something more virtuous than bullets. The smaller paper got named after the stone-tablet-shaped pills because it was compact and, well, "easier to swallow." Only worth mentioning here because some of the first tabloid-style newspapers were technically not tabloid-sized, but full broadsheets, printed on paper big enough to fold in half.)