Your example isn't a counterexample, because an invalid argument is one in which, in some model, the premisses are true, and yet the conclusion is false. Your conclusion is true. Indeed, if the conclusion is a tautology, then by definition it's always true, and there's no possibility of the argument being invalid.
For the second problem, I would say the answer is a contingent sentence. It's certainly not a tautology or a contradiction. It's a contingent sentence, because its truth depends on whether anyone has ever crossed the Rubicon.