Is there a client software (similar to DropBox) or any way to mount a disk to get access to the content of one's iCloud?
I'm talking iTunes-backups and other files synced to iCloud.
Not currently. The 'disk' space in iCloud is not like the iDisk option in MobileMe. Since iOS doesn't really expose its file system, they don't map this to the Mac in a 'mountable' way.
So currently, the only way to access iCloud data would be through APIs.