Directory /ETC has been moved! — Ubuntu 13.04

Boot from a live-CD and move it back to where it belongs.

Have you tried using a bootable medium (e.g. CD)? There are quite a few distributions out there that let you boot from an external medium, but let you mount your internal hard-drive. Knoppix is a tried and tested distro for these kinds of things, but there are plenty of alternatives.

Of course it's a bit of a catch-22 to ask you to download a copy of such a distro if you cannot even boot your computer. But perhaps you have a second machine available, or a friend could download a copy for you.

You will also have to configure your computer's BIOS so that it will actually boot from an external medium, rather than your built-in hard-drive, so check the boot options in your BIOS. Once your machine is fixed, you can reset that to the original configuration.

In GRUB, one of your options should be something with "rescue" in it. After booting it's likely going to drop you into an emergency sulogin root shell. From there you can copy it back and possibly even continue booting normally.

You probably can liveboot off your Ubuntu 13.04 cd as well.

Syncing the Linux running environment will involve bits and pieces of /var as well. However you should sync directories/files in /etc and /var "piecemeal" - i.e. don't do the whole directory but focus on programs whose configuration you want to make the same and then only sync those config files. Also Dropbox is probably a bad tool for this, you probably want to look into rsync or something similar.