If I were a golden fish, I would stick PostIt notes everywhere

To connect to a full remote X desktop using cygwin: go and copy /usr/X11R6/bin/startxdmcp.bat and change SET REMOTE_HOST=10.0.0.1 to match the server you try to connect to.
If all you get is a grey screen, ssh to your server go and set "Enable=true" in the [xdmcp] section of /etc/gdm/gdm.conf (if you're not using gdm, rtfm).