The first problem - libraries loaning out a Kindle - does not appear to be resolved. Again, it seems likely that market forces will eventually resolve this issue. Maybe.
The second problem - allowing a library patron to download a library eBook onto their own Kindle seems to have been resolved. Check out (pun intended) this article by Julie Bosman in the NY Times - Coming to Your Kindle: Library Books. The key factor - think "market forces" - appears to be summarized rather neatly in this simple observation by Bosman:
Amazon’s dedicated e-reader is not compatible with library e-books, leading many new e-reader buyers who are interested in borrowing e-books from the library to purchase a Nook from Barnes & Noble instead.How long will it take for Amazon to come to terms with the fact that readers will need to borrow not only the eBook but the device they need to read it?