Michael,
It would be good to get a baseline of what works. Perhaps you can try creating a layer package from a simple (small) data source and see if that works. It seems like the problem is with Desktop being able to create the layer package at all. If Desktop can't create a valid layer package, then the upload/download from ArcGIS.com won't make any difference.
Once uploaded, if you plan to share with a group, make sure you have the setting at the top of the application set to "Show: All Content" otherwise it will appear that the layer package is not getting shared with the group.
Thanks,
Mike