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.)

First note that the Enchanting page on the wiki says that Protection, Fire Protection, Blast Protection, and Projectile Protection are all mutually exclusive, but it looks like you can have Feather Falling on boots with one of the other enchantments.

The Armor page linked in the question has a table of "Enchantment Protection Factors" provided by each enchantment. I have duplicated the relevant parts of the table below:

Since increasing the level of the enchantment only increases the EPF, when trying to find the "best" protection it is only worth looking at the highest level enchantment. The article also says that in order to maximize protection from a particular kind of damage, you would need 25 points of protection for that damage type.

Since there are 3 different types of damage, one would need 75 effective points of protection, but since each armor can add either up to 11 points to one type of protection or 5 points to each type (which is like adding 15 total), the max is 60 effective points of protection. So, you can't max out everything. And plain Protection can't be maxed out at all. Therefore, the best armor depends on what you are looking for:

If you want to maximize the minimum protection value:

Put Protection IV on each piece of armor. This will give you 20 points in every type of protection. As a side bonus, if you put any level of Feather Falling on the boots, you will max out fall damage protection.

• Protection EPF: 20
• Fire EPF: 20
• Blast EPF: 20
• Projectile EPF: 20
• Feather Falling EPF: 38 (capped to 25)

If you want to maximize the average of protection values:

Again, put Protection IV on each piece of armor. Adding Protection IV to a piece of armor adds 5 to each protection type, so it adds 5 to the average. Adding any other protection enchantment adds at most 11/3<5 EPF on average.

• Protection EPF: 20
• Fire EPF: 20
• Blast EPF: 20
• Projectile EPF: 20
• Feather Falling EPF: 38 (capped to 25)

If you want to prioritize a particular protection type:

Plain old Protection:

Put Protection IV on all 4 pieces of armor.

• Protection EPF: 20
• Fire EPF: 20
• Blast EPF: 20
• Projectile EPF: 20
• Feather Falling EPF: 38 (capped to 25)

Projectile Protection then Blast Protection:

Put Projectile Protection IV on two pieces of armor, Blast Protection IV on one piece of armor, and Protection IV on the last piece of armor.

• Protection EPF: 5
• Fire EPF: 5
• Blast EPF: 16
• Projectile EPF: 27 (capped to 25)
• Feather Falling: 23

Projectile Protection, then Fire Protection:

Put Projectile Protection IV on two pieces of armor, Fire Protection IV on one piece of armor, and Protection IV on the last piece of armor.

• Protection EPF: 5
• Fire EPF: 14
• Blast EPF: 5
• Projectile EPF: 27 (capped to 25)
• Feather Falling: 23

Blast Protection then Projectile Protection:

Put Blast Protection IV on two pieces of armor, Projectile Protection IV on one piece of armor, and Protection IV on the last piece of armor.

• Protection EPF: 5
• Fire EPF: 5
• Blast EPF: 27 (capped to 25)
• Projectile EPF: 16
• Feather Falling: 23

Blast Protection then Fire Protection:

Put Blast Protection IV on two pieces of armor, Fire Protection IV on one piece of armor, and Protection IV on the last piece of armor.

• Protection EPF: 5
• Fire EPF: 14
• Blast EPF: 27 (capped to 25)
• Projectile EPF: 5
• Feather Falling: 23

Fire Protection, then the other two

Put Fire Protection IV on 2 pieces of armor and Protection IV on two pieces of armor.

• Protection EPF: 10
• Fire EPF: 28 (capped to 25)
• Blast EPF: 10
• Projectile EPF: 10
• Feather Falling: 28 (capped to 25)

Fire Protection then Projectile Protection:

Put Fire Protection IV on 3 pieces of armor and Projectile Protection IV on one piece of armor.

• Protection EPF: 10
• Fire EPF: 27 (capped to 25)
• Blast EPF: 0
• Projectile EPF: 11
• Feather Falling: 18

Fire Protection then Blast Protection:

Put Fire Protection IV on 3 pieces of armor and Blast Protection IV on one piece of armor.

• Protection EPF: 0
• Fire EPF: 27 (capped to 25)
• Blast EPF: 11
• Projectile EPF: 0
• Feather Falling: 18