Why is the root partition on my disk full?

One of my hypothesis is that when installing Ubuntu 12.04, I didn't configure my disks well and the disk /dev/sda6 is not mounted well as /home. Is this the reason indeed?

Yes!

What should I do to verify this

gparted clearly shows that /dev/sda6, although it contains some data, is NOT mounted as /home (or as anything).

and then to get the things fixed?

Outline: You would temporarily mount /dev/sda6, clean it up (or just format it, if the existing data doesn't matter), copy everything in your /home to it, unmount it and then set up /etc/fstab to mount it as /home from then on. Then reboot...

Official Guide on moving your home directory

Please follow this official guide which explains the process in more detail, and ask us any questions or concerns you may have in the comments.

As @jrg indicates in the comments though, if your /home is encrypted things get a little more complicated. Please ask a new question mentioning that in the title if you are in this situation.