Does deck choice (basic or constructed) have any impact on who you are matched with in Play mode

The fastest way to level is hands down by playing games against real opponents. This can be done in either play mode or Arena.

Arena may be best (if you're doing well for gold or don't mind spending the money) because you get to use cards you wouldn't have and the wins and losses still grant you experience. The arena cards you get to use would grant you a stronger deck for that class and would give you (a theoretically) high chance to win than with the basic decks you have. Please note that you will only have the option between 3 classes in arena and may not get the one you need to level (credit: ChrisHateZ)

You can still create custom decks even at low levels. Whatever common cards you may have earned can help in this cause so you can build stronger decks. In play mode, you can gain the experience in either casual or ranked. There is no experience difference gained there, just your ranking.

Also, games that last longer (in the same level) generally give more experience than quick ending games in my experience.

The only reasons to play against the expert AI is to test a deck you are building before putting it out into real play and to complete the quest that requires you to defeat all Expert AI opponents (which will reward you will gold).

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.