Well, that was fun.
I downloaded the caped model. When I brought it into blender with the standard FBX importer the armature was posed, but the mesh was not skinned and was in the default A pose. In Blender terminology, it had an armature modifier but no vertex groups or vertex weights. Strange.
Importing it with the Better FBX addon fixed the skinning issue but the armature and mesh were returned to the default A pose. So I used the built-in FBX importer to bring in a second, posed armature and copied the pose. It's all set now. If you want to send me your email via PM I can send you the blend, FBX, or OBJ file, whatever you want. (You'll have to sort out the textures, I didn't even look at that.)
BTW there were no problems importing the FBX into Maya. I wouldn't fault Blender's default FBX importer out of hand . . . there are many versions of FBX, Autodesk seems to tweak it a lot so it's probably not easy for third-party vendors to keep up. For exporting animation out of Blender I've stopped using it and use OBJ/MDD instead.