Hi,
so blockchain.info did switch to this new interface, right? I know it has been implented permanentely without the option to switch to legacy, like a few month ago. Back then I did download the Backup (which is a file named "wallet.aes.json"), after that I still received and spent a few BTCs.
So, does that file contain the private key (which I believe is necessary to import the BTC address to another wallet)? If yes, how do I exactly import my blockchain wallet to another wallet?
You have to decrypt the file (or some stuff in the file, I don't know exactly). This will decrypt the private keys. Then you take those private keys and import them into another wallet. How to import depends on the wallet.
And if I manage to import it, do I have the exact same amount I have currently on blockchain.info, as I said I did receive and spent a few BTCs after I made the backup in the old interface.
As long as no Bitcoin was spent between the time of the backup creation and when you imported, yes. The wallet backup does not actually store Bitcoin. The Bitcoin is "stored" on the blockchain. What the wallet saves is actually the information necessary to spend those Bitcoin (i.e. the private keys).
And can I recommend a desktop wallet which offers a two factor authentication (e.g. SMS code that is sent to your mobile device).
No desktop wallet has any such two factor authentication. This would require the use of a central third party service which no wallet developer wants their users to use as this could mean that the wallet would no longer function should that service shut down.
