1) She can get books in DRM-free Mobipocket/AZW format, or any DRM-free format which can be converted to Mobipocket (say, ePub via Calibre).
Examples may be found via Fictionwise's "MultiFormat" offerings and Baen's Webscription.net, which give you all the formats with a single purchase, and AllRomance eBooks, which only allows you to select one format, but has plenty of DRM-free Mobipocket offerings.
She should avoid "Secure Mobipocket", whose DRM the Kindle does not read, unless she's willing and capable of handling the tools to deal with it.
By the same token, she can get regular and password-protected PDF, but should avoid "Secure PDF" or "Secure Adobe PDF" which requires Adobe Digital Editions, because that DRM is incompatible with the Kindle.
2) Library books are a no-go, unless she's personally comfortable with stripping and breaking DRM and can handle the skills that go with it (fewer than it used to require, practically point and click nowadays).
She could legally get the (increasingly fewer) Secure Mobipocket books that libraries offer, but that still requires the use of a few tools to tweak the files so that the Kindle can read the old Mobi-DRM (this does not strip DRM; the books will still "expire" and become unreadable when the loan time is over).
3) In the US and UK at least, Amazon has a "free" category for its Bestsellers list. Otherwise there's a couple of sites which aggregate the time-limited free promo books. There's
eReader IQ,
Inkmesh,
Books on the Knob,
Randomize Me. The last two are blogs run by our very own MR members. And of course, the Deals forum here
4) Sure. She can pick any PRC/Mobi book from our Patricia Clark Memorial Library, and for that matter, any other format and convert it via Calibre.
Hope this helps.