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.
A win on your first turn is possible if you play as a Druid against another Druid who plays Millhouse on their first turn.
In the video below, the opposing druid opens with an Innervate and summons Millhouse Manastorm. Millhouse's battlecry says: "Enemy spells cost (0) next turn." (Which means your spells cost (0) on your opening turn.)
Next, the player uses this opportunity play just shy of their entire deck and win. A pivotal card is Gadgetzan Auctioneer, which gives them a card every time they play a spell - giving them a virtually endless supply of 0-mana spells.
Source
Best Answer
No, you'll just earn 6 cards (1 legendary and 5 'commons').
Cards you'll earn :
Haunted Creeper good for a zoo deck (it's a deck with low cost cards and very agressive) you'll earn this if you beat Anub'Rekhan in normal.
Nerub'ar Weblord good for counter battlecry, you'll earn this if you beat Faerlina in normal.
Nerubian Egg it'll combo with a solfurie or an Argus, because if you destroy it, you'll earn a nerubian (it's a 4/4 cards), you'll earn this if you beat Maexxna in normal.
Maexxna she destroy any card she attacked, it can be usefull if you play Anduin Wrynn because he can heal Maexxna. You'll earn this if you beat all the 3 boss in normal and you'll unlock heroic and druid and rogue class challenges.
Poison Seeds : destroy all monsters on the battlefield and give treants 2/2 for each destroyed monster. Can combo with a druid token. Unlock when you finish the druid class challenge.
Anub'ar Ambusher a 5/5 for 4 mana but with a Deathrattle (Return a random friendly monster in your hand) can combo with Si 7. Unlock when you finish the rogue class challenge.
You can find many others information on the wiki : http://hearthstone.gamepedia.com/Curse_of_Naxxramas