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.
So from the looks of things, there aren't actually many Easter Eggs in Hearthstone, at least not many discovered yet. It is indeed a little surprising, but to be fair, Hearthstone isn't exactly the largest of games. The closest things to Easter Eggs are actually more like hidden references to the lore of Warcraft.
- If Cairne Bloodhoof is played against a warrior, he says the line "Garrosh, you are not fit to rule the horde" rather than his usual line.
- If Illidan Stormrage is played against a druid, he says the line "Hello, brother" rather than his usual line.
- Millhouse Manastorm, the hero in the tutorial, says a line "Just you wait until I have 10 mana!" This is a reference to the old Millhouse Manastorm card, which had "Battlecry: Put a 'Mega-Blast' card in your hand." Megablast is a 10 mana card that deals 5 damage to all enemies. Reference
- In the Hearthstone credits, there are personalized cards for team members of Hearthstone, many of which seem to reference aspects of the individual person. Reference
- The opening video has an 8 cost Pyroblast; it's now 10.
- If you click on the vegetables long enough you can get a boot.
- If you play Dire Wolf Alpha against the Big Bad Wolf, your wolf whimpers after the Big Bad Wolf scared him. Reference Video
There are of course individual cards that are references to things in World of Warcraft rather than the actual lore, one of the most obvious being Leeroy Jenkins, but I wouldn't necessarily define these kinds of things as Easter Eggs.
If individual unlisted extras are discovered, feel free to edit this question and tack them on here, so they can be easily organized. However, from what my research could find, these are pretty much the only ones discovered thusfar.
Best Answer
This can happen if you haven't unlocked the Demon Hunter class but have previously completed the tutorial. Complete the Ashes of the Outland Prologue in Solo Adventures to unlock the Demon Hunter class and your extra deck slots again.