Prior to the passage of the 25th Amendment, when the Vice President became President, the office of Vice President was vacant until the next general election. This was last the case when Lyndon Johnson succeeded John F. Kennedy. In that instance, the Senate President Pro Tempore becomes its presiding officer and second behind the Speaker of the House in line to the Presidency should the office become vacant again. This has fortunately never happened; but there was genuine concern when Lyndon Johnson succeeded, as he had previously suffered a serious heart attack, and the Speaker of the House (Sam Rayburn) and President Pro Tempore of the Senate (Strom Thurmond) were both elderly. This was a primary consideration in the passage of the 25th Amendment which provides for a vacancy in the office of Vice President to be filled. As mentioned above, Gerald Ford became Vice President and later President under this Amendment. He is the only President of the United States who was never elected.
Most recently, this happened after Richard Nixon resigned the presidency in 1974. At that time Gerald Ford was vice president, so he became president and nominated Nelson Rockefeller for vice president.
Ironically, Ford himself became vice president via presidential nomination when Spiro Agnew resigned the vice presidency. So Ford served as vice president and president without being elected to either office.
Nothing really to add to #2, you've got it covered!
No, the vice president would not continue to fulfill his or her duties as vice president in that case. That person would then be the president. The 25th Amendment tells us what would happen in such a case. There would need to be a new vice president. This would be done by having the new president nominate a vice president to take that space. The nominee would have to be confirmed. Unlike Supreme Court justices, for example, who only have to be confirmed by the Senate, this new vice president would have to be confirmed by a majority vote of both houses of Congress.