Yes, Yes, and Yes.
As xzins has observed, what WOULD make a difference would be actual PROOF that Cruz's mother had formally (not by implication or inference) renounced her American citizenship BEFORE giving birth to Ted.
Yes, a document demonstrating Cruz’s mom had renounced here US citizenship or that she had received Canadian citizenship would be the only evidence I can think of that would take away Cruz’s citizenship. The first definitely would have. The 2nd probably would.