If MOE.exe is not running, then live mesh is not running. There could have been a problem with the installer that prevented it from finishing the install. As a good first step, check to see if the program shows up in the installed programs list in the control panel (Program and Features). If it is installed, time to check the registry for the autolaunch entry. I believe it is here:
There should be a key in there that points to "MOE.exe". If you then go to the install location, you can attempt to run the program manually.
I don't know if MOE would send errors to the event log, so it may be difficult to troubleshoot further. If all else fails, uninstall and reinstall the software. If it's an MSI file, you can use the "/log" switch to get detailed info on what is going on. I should warn you that the logging is not easy to follow for those who are unfamillier with it.
Hope this helps
Live mesh installs some files to your user profile. I personally think that's weird but here is where it is:
C:\Users[User Name]\AppData\Local\Microsoft\Live Mesh
To install, you should just be able to go to www.mesh.com and add a device to your account. It will give you the option to re-download the software there.