Answer to Question 1
The first step was measuring the distance to the Sun which then allowed us to calculate how luminous it is and therefore how much energy is needed to power it. The energy requirements are much larger than chemical reactions (i.e., fire) so this was then ruled out. A longer lived source that could match the energy requirements is gravitational contraction. However, as geologists and paleontologists found evidence for an ancient Earth, astronomers realized that gravitational collapse could not be the dominant energy source of our Sun today. All known energy sources were eliminated and only after the recognition that mass can be converted directly into energy, was the solution of the Sun's light as nuclear energy understood.
Answer to Question 2
It builds up there, for now the core is not hot enough to fuse it further.
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