I found solutions to two long-standing irritations in Cygwin on Greg Coombe’s website.

Cygwin exit gives error message

Does exiting Cygwin sometimes generate a “Terminate Batch Job(Y/N)” message? If so, you’re probably using a Windows batch file to launch Cygwin. If the batch file doesn’t consist of much, then just put all of that information directly into the shortcut. For me, this meant replacing the Cygwin shortcut:
C:\cygwin\cygwin.bat

to this:
C:\cygwin\bin\bash.exe –login -i

Cygwin: Can’t quit Emacs using C-x C-c

On some installations of Cygwin on Windows XP, it is impossible to exit Emacs using C-x C-x. After digging around in the message lists, I found out that the C-x key is trapped by the OS. The fix for this is to set a Windows environment variable:
CYGWIN=tty

Then quit all of the Cygwin windows and start them up again. This worked for me.