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Yes, they do. I believe it starts around 10% based on what I've heard from authors of not-tremendously-high-selling books. My friends who self-publish get much, much more. I believe digital sales/loans are also immune from "returns" or other chargebacks that apply to physical book sales.

I don't know how anything about getting a cut works, but my understanding is that authors don't have to allow their books to be in the library at all, and libraries specifically need to get permission to lend e-books.

Maybe just cynicism, but publishing houses are not known for generosity towards authors. (Much like music studios)

oh I'm sure they get /something/, its just not proportional to the their contribution.

I'm out of the habit of using the library because I was far away from one, but my normal approach is to just buy the physical copies of the book, regardless of whether I read that or the physical one. There are publishers who go out of their way to be drm free and i do make a point to buy those (Tor comes to mind).