In my experience running the game as admin isn't neccesary, provided the game is not installed into the Program Files folder. Also compatibility mode isn't required in my experience.
Running the game installer, the patch, the GWX installer and the SH3 Commander installer do need to be run as admin, as that is a requirement for a Windows installer program. Running (the latest versions of) JSGME seems it also requires to run as admin.
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