ad 1) There are currently three different mingw cross-compilers available:
The old gcc3 -mno-cygwin is deprecated.
There are new mingw64 cross-compilers for 32 bit (mingw64-i686-gcc) and 64 bit windows targets (mingw64-x86_64-gcc).

ad 2) There's no linux cross-compiler as cygwin package yet available.

ad 3) There's no cygwin cross-compiler as linux package available, but many mingw cross-compilers. Those mingw executables can also be executed under cygwin, though they cannot use cygwin features, just the simple windows runtime.

Correcting errors in others people posts:
-mswindows is not valid, -mwindows tells the linker to generate a GUI app without console.

ad 1) There are currently three different mingw cross-compilers available for cygwin:
The old gcc3 -mno-cygwin is deprecated.
There are new mingw64 cross-compilers for 32 bit (mingw64-i686-gcc) and 64 bit windows targets (mingw64-x86_64-gcc).
There's no mingw-i686-gcc matching the official cygwin gcc4 compiler yet.

ad 2) There's no linux cross-compiler as cygwin package yet available. But people report success building such a cross-compiler by themselves.

ad 3) There's no cygwin cross-compiler as linux package available, but many mingw cross-compilers. Those mingw executables can also be executed under cygwin, though they cannot use cygwin features, just the simple windows runtime.

Correcting errors in others people posts:
-mswindows is not valid, -mwindows tells the linker to generate a GUI app without console.
-mno-cygwin is only valid for the old deprecated gcc3 compiler and is not supported anymore. Don't use it. With cygwin you should use ordinary host and target triples as with every other cross-compiler.

ad 1) There are currently three different mingw cross-compilers available: The old gcc3 -mno-cygwin is deprecated. There are new mingw64 cross-compilers for 32 bit (mingw64-i686-gcc) and 64 bit windows targets (mingw64-x86_64-gcc).

ad 2) There's no linux cross-compiler as cygwin package yet available.

ad 3) There's no cygwin cross-compiler as linux package available, but many mingw cross-compilers. Those mingw executables can also be executed under cygwin, though they cannot use cygwin features, just the simple windows runtime.

ad 1) There are currently three different mingw cross-compilers available: 
The old gcc3 -mno-cygwin is deprecated. 
There are new mingw64 cross-compilers for 32 bit (mingw64-i686-gcc) and 64 bit windows targets (mingw64-x86_64-gcc).

ad 2) There's no linux cross-compiler as cygwin package yet available.

ad 3) There's no cygwin cross-compiler as linux package available, but many mingw cross-compilers. Those mingw executables can also be executed under cygwin, though they cannot use cygwin features, just the simple windows runtime.

Correcting errors in others people posts:
-mswindows is not valid, -mwindows tells the linker to generate a GUI app without console.