Truth be told, there are very few US Vice Presidents who did not leave office much better golfers than when they started.
I think Hillary will probably be offered the VP nod, but as she has struck me as being an all or nothing sort, she'll probably turn it down.
The concept of running mate goes back to the early days of the US. Both the President and Vice President had to be from the same party. Otherwise, if you had a Republican president and a Democrat vice president and the Congress was controlled by the Democrats, the Democrats might just cook up some phony charges, impeach the president and get their man in as president.
No references, that was the way it was explained to me.