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No. The President of the US will allays be referred to as President. It is normal for an ex-President to be called "Former President so and so" when introduced in a TV show, but the correct title is President until death. Same holds true for Judges and Senators, but not for Representatives or lower offices.
Yes a former US President can run for office of Vice President of the United States. A former US President can run and be elected for any other type of office.
Former US President Ronald Reagan
Monty Pythons FlyingCircus.
No former US president died in 1970. Closest was Dwight Eisenhower who died Mar 28, 1969.
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Martin Van Buren was the 8th US President.
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George H.W. Bush is the oldest living former US president. He is 86 years old.
the Commander in Chief Mr. President
The current President of the US is Barack Obama who was a former congressman from the state of Illinois.