There were a number of confirmation hearings in the Senate where Mike Pence, as President of the Senate, cast the tie-breaking vote. That part is clear to me from the Constitution.
Impeachment is also in the Constitution. The Senate shall have the sole Power to try all Impeachments . . . And no Person shall be convicted without the Concurrence of two thirds of the Members present.
Recently there have been some confirmation votes that were 50-49 and Pence was not required to vote which implies that just a majority of those voting were required for confirmation.
And recently there has been talk that 60 votes are required for funding a border wall. I think that is requirement due to Senate rules on budget measures—not something in the Constitution. Other than budgets, does anything else in the Senate require a cloture vote?
There was a noteworthy incident a while ago where all of the state senators left the state so that there wouldn’t be a quorum. Is a quorum required for anything in the US Senate?
So what are the rules for deciding votes in the Senate and where do they come from? Is it a simple majority of members or the majority of those voting? How does voting present impact the result? Are the rules the same for confirmation votes as for voting on bills?