1: What It Is!
It's a technical publishing term, meaning that the publisher of that edition -- a "book" is not out of print, an "edition" is, since a book can appear in an infinite number of editions -- has declared that it is out of print. It means that publisher does not intend to print or distribute that edition in the future.
It's kind of like declaring oneself a semiprozine; an edition is OOP because the publisher says so. There are many books that are not OOP, but the publisher doesn't have any copies in stock and doesn't plan to print any more -- those are OSI (Out of Stock Indefinitely) instead.
Many publishing contracts have clauses, particularly those concerned with reversion of rights to the author, that are triggered by an edition being OOP -- and that's one reason why some books are never declared OOP.
Editions have been known to come back from being OOP; there's no reason why they can't.
As a consumer, it's difficult to find out if a book is OOP, and it generally doesn't matter -- if a book is hard to find and/or expensive, its precise in-print status isn't important.
If you're looking to publish a book, the best way to find out what editions are in print (and thus what rights are available) is to query the author or (preferably) that author's agent.
2: Does "not being produced" mean that the publisher prints these books directly otherwise?
No, but there are commercial printers that do have printing plants which run pretty much all the time, printing books.
And the publishers have contracts with those plants, generally saying that the publisher will be using so much press time on these particular days, and their rates will be
Some publishers do, or used to, have their own "captive" printing plants, but that's generally not the most efficient way to operate, so it doesn't happen much anymore.
3: What about public domain books -- the kind Dover does?
Dover doesn't exclusively do public domain books, but they do prefer books that they don't have to pay for.
Public domain law is thorny and complicated. Here's a chart of the law in the US