According to this comment by @RickSmith:
"Congress may not 'ignore constitutional restraints or violate fundamental rights, and there should be a reasonable relation between the mode or method of proceeding established by the rule and the result which is sought to be attained. But within these limitations all matters of method are open to the determination of the house... The power to make rules is not one which once exercised is exhausted. It is a continuous power, always subject to be exercised by the house, and within the limitations suggested, absolute and beyond the challenge of any other body or tribunal.'" That the Constitution requires two-thirds to convict is a "constitutional restraint". That the vice president is President of the Senate (and therefore must be permitted to preside) is not, as I see it, a "constitutional restraint".
I would interpret the "within the limitations suggested" as invalidating the rest of the quote in the case of a violation of a "constitutional restraint" (the exact meaning of which is still arguable, more about that later).
First question: Does this mean that a blatant (or not) violation of the said "constitutional restraints" by the standing rules of the Senate would be subject to the challenge of another body or tribunal? (winking towards the Supreme Court)
(the special case of the impeachment of a sitting President, and the Chief Justice presiding, is set aside and ignored in all of the following)
Furthermore, Art. 1 Section 3 Clause 4 of the Constitution establishes the Vice-President of the United States as President of the Senate. Art. 1 Section 3 Clause 6 establishes a two-thirds majority requirement for conviction in an impeachment trial. I was asking on what grounds the former would be less of a "constitutional restraint" than the latter, and given that the current rules partially prohibit the VP from presiding in certain precise circumstances, if the same rules could overrule the two-thirds rule for impeachment trials. This lead to the question of the meaning of the VP being "President of the Senate". @Ricksmith said (emphasis his):
That the vice president is president of the Senate is a simple declarative that the vice president may preside when present. The Constitution does not require that the vice president preside...
The Constitution does indeed not require the VP to preside, but the rest seems more blurred. If presiding the Senate is not a constitutional right/power of the Vice-President, then one can only wonder what the "President of the Senate" part of that clause is for. And if not, if it implies for anyone not thus authorized by the Constitution (and not being a Senator) to be constitutionally forbidden from presiding over the Senate. That would mean that a Senate rule authorizing a non-senator to preside in a random circumstance would be unconstitutional, but a rule making a non-member preside is not unheard of in parliaments: it is commonly accepted that the House may elect a non-representative as Speaker, and in the UK during the mid-20th century, a random clerk used to preside over the House of Commons in specific circumstances (the election of a speaker). So I would simplify this part as:
Second question: Does the Constitution grant the Vice-President a constitutional right/power to preside over the Senate? If not, does it rather forbid anyone non-VP (and non-senator) from presiding, even if a Senate Rule were to allow it? If neither, then is "President of the Senate" just an empty title, does it not imply anything about the right of the VP to preside over the Senate?