Presidential succession of the president of the senate?

Answer 1

The President of the US Senate is the Vice President of the United States. Only someone holding the Office of the Vice President (NOT even the Acting Vice President) can occupy the President of the Senate.

If the V.P. is not currently present in the Senate, the position of President Pro Tempore of the Senate will act in his stead, performing the normal actions that the President of the Senate does, EXCEPTING that the PPT may not cast any tiebreaking vote - that is solely for the V.P. to do.

So, to answer your question, there is no succession for the President of the Senate. If the Office of the V.P. is vacant, then no one assumes its powers until a new V.P. is appointed and confirmed. That is, the powers of the Office of the V.P. do not transfer to anyone should it be vacant, and no one automatically fills the V.P. position on its vacancy.