Your symmetry indicator (L/R or Left/Right) hast to be between two separators like dots or dashes.

To make sure, that vertex groups are connected with bones, the action button only shows, when the active mesh has a armature as a direct parent, because I assumed it to be the most common way regarding rigging/skinning (have a look at my show case .blend file).

No, if you use the mirroring in vertex group, the values of the group will be transfered to verts on the other side, but stay in the same vertex group. The bone on the other side though, is controlled with a different bone and vertex group. To get the desired effect for weight painting, you would want to use the x-symmetrie function. But this will only cover your future strokes. My Add-On helps afterwards, when you forgot to use the symmetry and takes care of the mentioned conversion to the corresponding Bone on the other side. As far as I know, there is no similar funktion in Blender so far...

Yes, for the addon to work properly, the object has to be symmetrical. This also means that it has to be only one object in the context of the symmetrical armature.