Can I install linux on external hard-drive of my notebook
 

My notebook hard-drive is short of space, I bought external hard-drive, can I install linux on it to dual boot with my Windows OS ?

I'm not really up-to-date with grub development, but I think the easiest thing to do is making a 20MB partition on your disk to store kernel+modules, /bin, /sbin, /boot and the basic /etc, then mount /usr, /home, /var+tmp and swap on the external disk. To save space make /etc/X11 a symlink to /usr/etc/X11.