Question: When you play with a custom deck, you know every card that can show up, since you put each one of them in that deck. However, when you just pick a hero and play with his or her basic deck, you have anywhere between 10 and 20 basic cards that you can be sure of, depending on your progress with that hero, and the rest is made up from random other cards you own. Is there any way to find out which other cards are included? Are they all neutral, or can cards of your class occasionally fall in that mix? Are they picked randomly, or are there specific lists? Do different classes get different extra cards?

Answer: 
The basic deck attached to each character contains only free cards (neutral cards without gems) and the first 10 cards given to that class. These decks are always the same, and don't change when you receive additional basic class cards (while gaining levels) or open expert packs. 

Their purpose is to provide a simple deck to start out with, or to play against someone who is just starting out.
Resorath has answered the other stuff.

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> Do different classes get different extra cards?
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Each class has a number of unique cards to that specific class. Example: Fireball and Pyroblast. These cards are unique to the mage class and there are many more cards the mage has that are unique to mages only.

A helpful site to review every card in the game, make decks, compare decks, view decks, etc... [www.Hearthpwn.com](http://www.Hearthpwn.com)

Additionally if you would rather review the cards in-game you can go into crafting mode. Cards you do not have will appear, but will be transparent, and unusable in your non-arena decks. This is helpful because you can still click them to review the card and see the cost in arcane dust or how much you will get if you already have the card and would like to disenchant it.