We are told canon is what happens on screen.
Again, no. The canon is the whole work.
Star Trek is a work that consists, in its original form, of TV shows and movies. That's all that meant by "the canon is onscreen." It's not a literal statement of cause and effect, merely a clarification that
Star Trek books, comics and games, and the contents of interviews or behind-the-scenes production materials, are not part of the canon defined by the TV shows and movies.
Now how showrunners have treated canon has varied. The current writers have indicated some of the novels, particularly the Discovery novels (and I assume comics) have canon value...."until they don't" I think was the quote.
That is a bit different from prior regimes like the Berman regime. Berman's regime was much more concerned with continuity and on screen canon being the only source. It's just a change in focus.
It's not that great a difference. Berman-era Trek did draw on the novels at least once, adopting the Klingon Day of Honor concept invented for a novel crossover and basing an episode on it. And the movies did incorporate characters' first names invented for the novels.
The thing is, the creators of fiction aren't overly concerned with defining what is or isn't "real," because none of it is real. They're just telling stories. Past continuity, ideas from novels or comics, it's all just concepts that can be drawn on as raw materials for building stories. They don't have any time to waste building walls between one set of ideas and the next; that's something Roddenberry had time to be preoccupied with late in life because he no longer had a hands-on role on the show, and it's something that fans have the luxury to obsess over because they don't have to do the actual work of creating a TV series. What the producers of a show have to focus on is telling the actual stories, putting things onto the screen. Anything else is just a means to that end, and if they're able to pay attention to tie-ins and borrow ideas from them, then that's fine, but they usually can't be bothered to worry about staying absolutely consistent with the tie-ins, because they're just too busy.
The main reason they're paying attention to the novels now is because
Voyager novelist Kirsten Beyer got a job on the writing staff. The fact that she's worked on the novels means she gives them a new resource they can use, and they're taking advantage of that. But if they do borrow any ideas, they'll adapt them to fit the needs of the shows, rather than adapting the shows to fit the prior continuity of the novels.
No, it didn't. "Affliction"/"Divergence" established that the population of Klingons who lost their forehead ridges numbered only in the millions, a small fraction of the entire Klingon population. So it's easy enough to assume they were just not involved in the events of DSC. Somewhere in between DSC and TOS, policies must have changed in a way that put more smooth-headed Klingons on the front lines with the Federation, at least for about 5 years.