does every graph have a complete, preferred, stable and grounded extension?
As a complete extension is an admissible set containing exactly these arguments it defends, there is always at least one complete extension, that may be the empty set and nothing else. 

Grounded extension is a minimal complete extension, hence there is always one, and it also has the property to be a unique fixed point. 

Perferred extension are maximal complete ones, hence starting from the same principle (there is at least one complete extension) there should always be at least one. 

As far as I know, the notion of stable extension is not so well-defined and their existence therefore depends on the chosen definition.

You can probably work out additional properties from the paper 

Baroni, P., Caminada, M., &amp; Giacomin, M. (2011). An introduction to argumentation semantics. The Knowledge Engineering Review, 26(04), 365-410.