If A President Elected Dies Before Taking Office Does His Or Her VP Pick Become The Next President?

I believe that the VP Elect is sworn in as Pres. And the Majority Speaker of the House steps up to VP.

It depends on whether or not the electoral college has already made their decision.
Here is one analysis from Wik:

In the United States, the members of the U.S. Electoral College are elected by the people in November once every four years; in December, they are in session and in turn elect the President of the United States; finally, the President of the United States assumes office in January. One is officially the president-elect only after being chosen by the Electoral College, but unofficially the person chosen in the November general election is called the President-elect even before the Electoral College meets. An example of the practical effect of the official status is found in the U.S. Constitution's provision that if the President-elect dies, then the Vice President-elect becomes president on Inauguration Day. That rule takes effect only after the meeting of the Electoral College. If the person unofficially called the President-elect dies before that meeting, then the Electoral College would have broad discretion to choose some other person.