2. If you do not enter anything for a field type, nothing will be shown. You can test this out using any of the templates that come with Connections.
3. They can not be removed, but as mention in 2., if you do not enter any messenger IDs, nothing will be shown.
4. When entering data in a fields type, you can set the visibility to Unlisted, only the admins will be able to view that info. This too can be tested with the templates that come with Connections.
I hope that answers your questions thoroughly.