I've done a bit of playing about with it (the mail app anyway) and have it working in ModernMix mode (the live tile but not while the mail app is open).
There are two processes in the task manger when ModernMix is running - 1. Under 'Apps' that bundles all current open apps and 2. Under 'background processes'.
End both of them (unfortunately ending 1. closes all current apps)
Confirm that your mail app tile is now working.
Go into the ModernMix Program File and open 'MMix_32.exe'. Now open any app (or do vice versa - then the open app appears on task bar)
Finally go back in to task manager and end that ModernMix task under 'background processes'
I now can open up apps in ModernMix and my mail app tile is alerting me to new emails. The whole thing seems to break though if you reopen the mail app. That's alright for me, I just want the mail app to notify of new mail on start screen and lock screen. This is no use if you actually use the mail app. I would suggest installing the gmail touch app, set up your email to hook to gmail.
*Some* apps not opening in ModernMix mode though by doing that (small minority for me anyway, had to try a good few to see it happening), presumably because I shut that 'background' process. I'm willing to live with that though, I really need/want that mail app tile working on the lock screen because I have it on during work.