How do I set up SSH to open a single application? iMac 7,1 running OSX 10.10.3
Dell Inspiron 740, running Windows 7 Home Premium

My aging mother has a Windows PC and a Mac, which she prefers using. Sometimes my dad is using the Mac, so she's stuck using the Windows PC (it's specifically for her) until he's done. Her favorite part of the Mac is using the Mail app, which I've tried time and again to find an equivalent, but nothing's got everything. I've now had the thought to just set up PuTTY on her computer to open an Xsession via SSH, on a user specific to her, and have the Mail app open automatically. How do I set this up?
XQuartz, which is the OS X X-server implementation doesn't support native Aqua apps like Mail.app, just regular unix X-server apps - xterm, etc, so I don't think this will work for you.

However, OS X does come with VNC built in, if you want to share screens from Windows. It's under Sharing in System Preferences, check the box for 'Screen Sharing'. See this: Several users simultaneously on a Mac mini