What is an enchanted book?
According to the Minecraft Wiki, an enchanted book is an item which allows players to add enchantments to weapons, tools, or armor by combining the enchantment book with one of said items on an anvil.
Can they receive any enchantment?
Testing in creative and survival mode shows that books can receive any enchantment that a normal weapon/armor/tool/other can receive. An easy way to check is to go into creative mode and use the new search function to search for the term book. This will show you what all the enchanted books are.
Can they receive more than one enchantment?
According to the wiki, books may only have one enchantment. This can be tested by giving yourself a bunch of books and enchanting them, then throwing them into lava or whatever you wish to do with them.
Since enchantments have different chances of occurring for different tools, materials, and at different levels, how are those probabilities different for books?
This would be different for books since books are not diamond swords. However, this would have that many variables because unlike a sword, books can only be made from paper. Swords can be made from diamonds, iron, and other materials. You would have to test this to truly know, but if John tells us how many levels he invested into his enchanted books, we might be able to know. Right now, though, it appears that the enchantment gives you the highest level (ex. Protection IV, Sharpness IV, etc.)
If you're looking for something specific, you have to enchant a lot more books to get it than enchanting a specific tool, since there are more possibilities.
If you enchant a book at level 30, your chance of getting Looting II or III is approximately .02 (1 in 50), and so you will have to enchant about 50 books before you get it. If you have XP and time to burn, this is the way to go as books can be made from easy farming.
On the other hand, if you enchant a diamond sword at level 30, you have a .15 (1 in 7) chance of getting Looting II or III. (This should actually be even better because I'm not working in the effect of multiple enchantments.) Thus, you will get it much sooner — but you will also get (on average) six swords that don't have Looting on them. Note that you can then recycle those swords by repair-crafting them with almost-used-up diamond swords.
Therefore, given that you say you want to spend diamonds rather than XP, you should definitely enchant swords.
(all numbers from minecraftenchantmentcalculator, back when it existed, which is a sloppy tool because it just does a simulation rather than calculating true probabilities, but it was handy and should be roughly correct)
Best Answer
I do not know if there is any set documentation on the various times/efficiency for getting enchanted books, but let's take a look from gameplay experience.
This is not likely to be the fastest way, as fishing is meant for, well, fish. According to this, a vanilla Fishing Rod only has a 5% chance of catching a treasure, and from that 5% chance, there is only a 0.8% chance that it will be an Enchanted Book. So, your odds aren't so great.
Dungeon chests have a 15.1% chance of spawning an Enchanted Book. As the frequency of dungeons is already limited, again, this method is not so great.
This could be the best way, although other auto-fishing methods could work. By setting up a mob farm to generate XP, you can easily enchant multiple books at the highest possible level to do so. There are various YouTube videos and guides on setting up efficient mob farms, so that may be beneficial to check out.