Both are in common usage. The US Constitution uses the hyphen as well as capitalization: the Vice-President . Purists might say the hyphen is needed to avoid confusion with a possible president of...
The former vice-president, now the president, would nominate a new vice-president who has to be approved by both houses of Congress in order to take office.
The President Pro Tempore fills in when the vp and the president arnt there. If the president dies then they have the president choose another since the president chooses his vice president.