This may turn out to be one of those learning experiences where the final solution is to re-install.
The command you ran to try and fix things, chown -R root var only changed all the owners to root. John_VV suggested using chown -R root:root /var, to also catch the groups. This may get the majority of files & directories back in line, but there are quite a few files & directories that do not have an owner:group of root.
The only solution I can think of involves comparing the files and directories in /var with a proper installation. It might be possible to get most of it corrected using a Fedora Live CD, but ultimately there will be some that just don't match up. Although you could automate some of this using scripts it is still going to be rather time consuming.
This is only the tip of the /var iceberg, but it may get you going. gdm is where your mouse and keyboard are acting up so we'll concentrate on those entries.
.ICEauthority has the following characteristics.
-rw-------. 1 gdm gdm 18950 2009-10-29 08:00 /var/lib/gdm/.ICEauthority
The following directories and their contents all have gdm:gdm for their owner:group - /var/run/gdm, /var/lib/gdm, /var/log/gdm, /var/gdm.