Is every six-pointed star forbidden (haram)? And does the design of six-pointed stars on a prayer rug refer to the Star of David? And is its ruling similar to that of the cross? And if it is not permissible, what should one do with this rug?

Upon reviewing the engraving, it did not appear that the hexagonal shapes in the prayer rug symbolize the emblem of the Jewish state. Therefore, there is no harm in using it as is. Not every hexagonal shape symbolizes that, and one should not extensively entertain doubts.