Sorry about the issue you have with the add-on.
I do not use Unity3d so I cant test your issues. It sounds to me that the first issue is with the exporter’s defaults. That is why I implemented
Custom Options in the add-ons latest version.
You can add your own comma separated custom export options in there to export with. The options are based on the internal file exporter’s own options. You can hover over the actual FBX exporter in the Blender exporter UI and see what the option is called and then you can paste them there. Also there is a text file in the add-ons folder with those options.
See this image for how to format them.
I am also planning profile based exporting like Unity, Unreal etc. Please feel free to provide some good FBX defaults for Unity3d.
As far as the armature goes, you are right. This is an object based exporter meaning that it is meant to be exporting resulting meshes. And it is for repeated mass exports and imports. However I added armatures to my to do list. I can see how useful this for game developers.
I am thinking to add scene based exports (vs object based now). This should solve Fbx with armatures issue. But you can try the custom options to see if it works for you.