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: 
I did some research of my own instead of waiting for another answer. I played 4 games with my Level 5 Mage (leveling her up to level 8 in the process). In none of the games did I manage to use the whole deck, but I wrote down all the cards that I drew throughout all 4 games.

First, to confirm Resorath's answer, none of the Level 2, 4, and 6 cards of the Mage ever showed up in the deck, but the original 5 cards that the Mage starts with all showed up in at least 3 games out of 4. So I can conclude that none of the later Mage cards are included into the basic deck as you level up.

Second, exactly ten other cards showed up in 2 to 4 games each out of 4, while no other cards ever showed up. So, only 15 different cards ever showed up for the Mage, some twice, making up the deck of 30. That means, it's not random, and the cards are set in for each basic deck. Most likely, they are tailored to each class. (For the curious, for Mage they are: Murloc Raider, Bloodfen Raptor, Novice Engineer, River Crocolisk, Raid Leader, Wolfrider, Oasis Snapjaw, Sen'jin Shieldmasta, Nightblade, and Boulderfist Ogre, plus the original 5 Mage cards)

Both of those conclusions could be very good coincidences, but, eyeballing the probabilities of these coincidences, I firmly believe that those conclusions are correct.

So, I did not experiment with all 9 classes, but I found my answer: The basic decks were pre-made by the creators of the game, and they do not change as you level up. I guess the results for the other 8 can be found the same way.
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.