Depending on your politics the last Rí na hÉireann was either Ruaidhrí Ua Conchobhair (died. 1186) or Edward VIII of the United Kingdom (reigned. 1936).
Oh dear, is the descendant of Ruaidhrí Ua Conchobhair most pretendant to the throne not known? Most of the houses of republican western Europe still have one.
The position of High King of Ireland was not a hereditary role - Ruaidhrí Ua Conchobhair didn't succeed from his father, and only became High King 10 years after his fathers death, as a result of conquest. His descendants have no more legitimate claim to the title than anyone else.
Because "Irish" is an adjective, and there is no adjective in Rí na hÉireann.
The short answer is yes, but there is a more involved answer.
In English, "The King of X" is a rank or title, "the Xish King" is just a description. It often matters which form you use (not always, but you can't simply ignore the difference).
In other phrases, such as rialtas na hÉireann or Bunreacht na hÉireann, the translations "the government of Ireland" and "the Irish government" or "the Constitution of Ireland" and "the Irish Constitution" are equally acceptable.