Yes, although it hasn't happened since 1976, when President Ford's running mate was Bob Dole instead of incumbent Vice President Nelson Rockefeller, and before that it hadn't happened since 1944, when President Roosevelt's running mate was Harry Truman instead of incumbent Vice President Henry Wallace.
Bill Clinton is not eligible to be vice-president since he has already been twice elected as president.
The short answer is a president runs an entire country, where a governor just runs one state. Thus, a president may sign legislation that affects the states, and has far more authority than a governor does.
The president and the vice president runs the office together as a unit
Only people that runs for president and not everyone who runs for it will get it
The Vice President runs in tandem with the President and is of the same political party. Either or both houses of Congress can be controlled by an opposing party and they choose their own leaders. The Vice President is the presiding officer of the US Senate and can vote in the case of a tie .
The Vice President. If the Vice President is absent, the Secretary runs the meeting.
The Prime Minister officially runs the day-to-day affairs and runs the government. the President is head of State, but not of the government (which can be of a different political party).
The presidential candidate really doesn't choose the person he runs with. He/she will choose a person who brings something to the ticket they need. It could be electors, experience, connections. If a brother or sister was within this format they could run with them, but the party convention votes for a ticket and not each person. In the case of JFK his brother Robert became Attorney General of the United States.
president
The president runs the Executive Branch
If he has been elected President only once, he usually runs for President.
The person who runs Austria is the president Heinz Fischer