Xayma is correct in pointing out that styrene is retained for C<sub>6</sub>H<sub>5</sub>-CH=CH<sub>2</sub> according to rule A-12. Ethenylbenzene is the correct fully systematic name with benzene as the parent chain, although vinylbenzene is also acceptable. Taking the ethene as the main chain, the name would be phenylethene.
I don't what 'vinil-phenyl' is supposed to be, but it sounds wrong to me.
On the subject of benzene, C<sub>6</sub>H<sub>6</sub>, it has no other name. It should not be referred to as 1,3,5-cyclohexatriene, for two reasons. Firstly, because 1,3,5-cyclohexatriene does not exist. Secondly, because this name is reserved for the non-existent 6 membered cyclic system with alternating single and double bonds. The stability of benzene is usually measured relative to this hypothetical model, and so this name is not used for benzene itself.