The limitations of the "qualifications" power of the House were established in the Adam Clayton Powell case. To my knowledge, there has been no such court ruling on the power to judge elections. Presumably, if acting as "judge," the Senate would need some rationale for its decision; it lacks the power to appoint Senators, but if there are two plausible vote counts, the Senate can choose either.
see #19
But who gets to define the word "plausible"?