Does Gödel's incompleteness theorem invoke a Law of Excluded Middle contradiction? Does Gödel's incompleteness theorem cause the Law of Non Contradiction to contradict its self? If so, would this be a considered a conjecture?

No, not really. Gödel's theorem does not appeal to notion of "truth" of a conjecture _per se_ \- merely to its provability within a given set of axioms. The "provability" operator can be intuitively understood as $\square_A \psi= A \Rightarrow \psi$, $A$ being the set of axioms: even under the law of excluded middle, it is not guaranteed that $[\lnot(\xi \Rightarrow \psi)] \Rightarrow (\xi \Rightarrow \lnot \psi)$.