[The Squatter of Amon Rûdh]

the little of Mythology England had, was actualy composed of other cultures myths such as "King Arthur" which actualy has a french origin

Actually it's Celtic. The French elements, such as Camelot and Lancelot were added by Chrétien de Troyes at a much later date. Tolkien's objection to the Arthurian legends was two-fold: they were not English (Anglo-Saxon) in origin, and they had been hopelessly bastardised over time.

i'm also interested of your opinion of my idea that chess could have been brought to western middle-earth by the Bluew wizards.

There's no reason why it shouldn't have been. I do think, however, that the evidence is a bit scanty for an argument one way or the other.