I'd be really grateful if someone could point me to a FAQ on how to cross compile mint kernel on Cygwin.
I've tried to figure out this myself, but sadly the last time I've tried to build my kernel was about 10 years ago.
Nothing hard, really. You need to have mintlib installed (for a few headers), you can safely bet on Vincent's work here. For XaAES & tools you need gemlib & cflib installed. In general, it's always about 'make CROSS=yes' and making sure you have all needed libraries (what is quite easy to figure out from make outputs). Kernel is build in exactly the same way, you might want to disable some targets in KERNELDEFS and XAAESDEFS files.