I don't know what the defaults are for cygwin's SSH.EXE, but for openssh the default is to not enable X11 forwarding.
That default can be overridden by modifying the ssh client's config file (e.g. ~/.ssh/config on a unix/linux box) or by using the -X option on the ssh command line - e.g. ssh -X remotehost gimp
Might be worthwhile checking whether cygwin SSH.exe has the same default and/or the same or similar option.
BTW, what happens when you ssh to the Mint box and then run gimp from the command line? if it doesn't work, try again with -X.
Finally, you may want to try putty as your ssh client on the windows box.
You can try to use VNC X server. It uses non-privileged port to communicate and it may be run without any root privileges. To avoid the building of VNC find out what port of it the distro being in use contains (there is a number of options TigerVNC, OpenVNC, RealVNC, e.t.c.).
For example the Fedora 17 has tigervnc-server-minimal package that has everything you need to start a VNC server:
/usr/bin/Xvnc
/usr/bin/vncconfig
/usr/bin/vncpasswd
/usr/share/man/man1/Xvnc.1.gz
/usr/share/man/man1/vncconfig.1.gz
/usr/share/man/man1/vncpasswd.1.gz
Download it, extract the binaries and put them into your ~/bin folder for convenience.
First you need to run vncpasswd once at each system to set a password to access your vnc server instance.
Then start the server itself by the command Xvnc and note what display it started (it will print out the info on standard output).
Then you will be setting up a TCP port forwarding with putty to the port with number 5900+<display number>, e.g. for the display :1 you should create a tunnel to port 5901:
putty -ssh -L5901:127.0.0.1:5901 user@host
Then start the VncViewer and connect to the display localhost:1 at your Windows box.
When you are finished don't forget to stop Xvnc server, so it is not wasting the resources at server:
killall Xvnc
The case of aura is a bit more complex as you can't log in directly. If one of your servers allow to set the tunnels to any machine in the LAN, then just create the proper tunnel, say:
putty -ssh -L5901:<ip-of-aura>:5901 user@host
Otherwise, you start the ssh session with aura with port forwarding from the remote shell at aisa or lethe:
ssh -L5901:127.0.0.1:5901 aura
Best Answer
Depends on the Windows program, but generally, no.
The reason those linux programs can throw up their display on a PC is because they are written for the X Window System, which completely separates the client from the display server.
X has been ported to virtually every system out there, and is the defacto standard for grpahical programs on Unix/Linux variants. More specifically, any program that linked against Xlib would work in the other direction just fine. So if you were running, say, GNU Emacs in a Cygwin/X environment on Windows, you could put that program's display on Linux no problem.
But generally, no: your classic win32 programs (say, anything that ships with Windows, or Office, or your web browser, games, etc) are not going to be able to ship their display to an X Server, because they are not using Xlib at all.
What you can do is run an RDP client to let you log into the Windows desktop and run a full desktop session (but admittedly, that's quite a different solution that displaying individual programs).