Transaction history size I recently installed Bitcoin-Qt on my laptop and it started performing the synchronisation with the network since a couple of weeks. And it's not finished.

Why do the Bitcoin-Qt application have to do so? If I am right, this application is just a Bitcoin client... I mean, it have not to check transaction integrity such as a miner, right?

Also, I would know if according to you, in 10 years (with some imaginations), having a lot of transaction every days, this synchronisation could take some months or even more.

Thanks for your lights!
The main reason to run Bitcoin-Qt is to help the bitcoin network by relaying transactions and blocks as well as verifying the blockchain.  Bitcoin is a distributed protocol and relies on its users for validation.  You can download an initial blockchain so that your client doesn't need to get it from a peer node (your download will be limited by the upload bandwidth of the peer node).  Once synchronized, the network load is around 144 MBytes/day depending on the number of connections and the needs of your peer nodes (your upload bandwidth can be saturated for hours if another node needs to download the blockchain from you).

That said, if you just want to use bitcoins without participating in the network, you should use a light client such as Multibit, Electrum, or Armory.  Check here for more information.