Startxwin is a shell script (/usr/bin/startxwin). It creates the magic cookie file (to prevent from unauthorized programs from taking over your screen), and then does a normal "xinit". Here's the command it uses to start a server on my box:
eval xinit "/etc/X11/xinit/startxwinrc" -- "/usr/bin/XWin" :0 -multiwindow -nolisten tcp -auth '/home/MY_USERNAME/.serverauth.1648'
The startxwinrc tries to spawn "something" that has the cookie and display stuff set for you.
It also takes many parameters: type 'man startxwin' for help.
But, for your shortest and simplest answer, create a shortcut with the following command, and it will start your X server, without any output or extra windows:
C:\cygwin\bin\run.exe -p /usr/X11R6/bin XWin -multiwindow -clipboard -silent-dup-error
That should set you up. (If you installed cygwin somewhere other than c:\cygwin, just fix the first part of that shortcut)