There's no icon of dropbox in the gnome-shell's tray (bottom-right corner). There's just text. I tried different icon themes but still it's missing. Here's a screenshot:
How did you install dropbox?
The following should work fine (including in gnome-shell):
When asked to install dropbox from the original location choose not to. After installation it will ask for your login details.
You can start it manually (from a terminal) with: