For 1) what is your definition of transitive? I aks, because there is more than one definition, and if you were to take the definition `for all there exists a such that ' then your result follows quite easily. In fact, I would suggest proving that your definition and the one I just gave are equivalent, and proving the result that way.

For 2) prove that the orbit containing x has even order (i.e. show that if is in the orbit then so is , and that ), and then apply the orbit-stabiliser theorem.