Hello,
thank you for using Trombinoscope Extended (TE). For your concerns:

1) You chose Theme 1 which is made to have a margin starting a the third field. If you do not want this margin, choose the 'Original' theme or you will have to override the theme 1's CSS like so:
2) In order to have Facebook or other Social link, your contacts need to have the links into their contact sheet (Joomla! contact component, links A through E). Then you need to select those links in the TE component (Link fields). TE will automatically detect the social links and show the proper icons.

3) The category label in TE links to the contact category Joomla! component. That means: all necessary changes of that page need to be made through the Joomla! contact category options. If you need a different look, you will need to go through template overrides.

Glad it helped.
There is no way for the user to update the links a through e at this time. The component is not a full blown social component like JomSocial is, it is mainly meant to show contact information in a different way (think of it as a fancy contact category view). I have been trying to address more issues over the months but some are still under work.