It wasn't sarcasm, I do not know that it is impossible. If you do get it working, I may
be interested in setting up a similar thing on my PC. Now that you mention it though,
I do remember reading something from the cygwin site that made me THINK it was
impossible (once upon a time).

Code:
Cygwin is not: 
•a way to run native Linux apps on Windows. You must rebuild your application from source if you want it to run on Windows.
So doesn't that mean you would need to build the linux version of ADOM from source?
Who knows, maybe sage will take care of all the 'making linux-ADOM run' stuff.

I am also interested in how much HD and RAM resources that cygwin takes up
to get as far along as you have gotten?