(1) "The European summer of singer and a dancer James Green"
(2) "The European summer of the singer and dancer James Green"
(3) "The European summer of a singer and a dancer, James Green"
They are all okay, though it may be helpful to consider the syntax:
In (1) and (2) the composite NP "(the) singer and dancer" is head of the underlined NP. The noun "James Green" is called an "appositive" modifier; it can replace "(the) singer and dancer" to yield an entailment of the original NP, i.e. "The European summer of James Green". The article "the" serves to mark the phrase as 'definite', which here implies that the reader or listener is already familiar with the person named "James Green". The omission of "the" in (1) is typical journalese; only marginally ungrammatical if at all.
In (3) things are somewhat different. Although "James Green" is again an appositive, it is of the supplementary (non-defining) kind - note the insertion of a comma after "dancer". And unlike in (2) it is no longer a modifier but a separate unit of information, parenthetical in nature, set apart by the comma, and by a slight pause in speech. And this time the article "a" marks the phrase as indefinite; it assumes that the reader or listener is unfamiliar with "James Green".