To run an X server on Windows under Cygwin, use
C:\cygwin\bin\XWin.exeThis can be run as a scheduled task in Windows using
Start/Control Panel/System and Maintenance/Administrative Tools/Schedule tasksThe basic program starts a big block window which covers your window screen, without a window manager. You can use the twm window manager.
If you want your X windows to share the screen with Windows windows, you can run the X server with the option
C:\cygwin\bin\XWin.exe -multiwindow
This does not allow cut and paste. If you want to also be able to cut and paste between the X windows and the Windows windows, there is an option -clipboard
C:\cygwin\bin\XWin.exe -multiwindow -clipboardwhich makes it possible.
The -clipboard option also alters the behaviour of the keyboard. My keyboard is a Japanese model, so I use the following two options
C:\cygwin\bin\XWin.exe -multiwindow -clipboard -xkblayout jp -xkbmodel jp106This gives correct correspondence between the keyboard and the output on the X Windows windows.