Hi Matthias,
if I understand your question correctly, you would like to define a structure rule for all cards of type "Contact".
I think the easiest way is to go to the "Contact" card and choose from the menu advanced -> rules -> All "Contact" cards.
Make your changes there in the "structure" section (eg. add {{+phone}, {{+name}}, etc } and choose as set "All Contact cards".
Some technical details that might help to clarify your problems:
There is a difference between defining a structure for the card "Contact" and a structure for all cards of card type "Contact".
The former is stored in "Contact+*structure", the latter in "Contact+*type+*structure". So an alternative solution is to edit these cards directly.
Because the card "Contact" is of type "Cardtype" there is already a default structure for this card defined in "Cardtype+*type+*structure" (it adds the description field you came across and a list of all cards of that type).
At your point 2) when you had a look at the edit menu of "Contact" there should have been the option to edit this structure but it would have been the wrong structure anyway.
Hope this helps,
Philipp