The core temperature of the sun is on the order of 15 million degrees kelvin while its surface temperature is around 6000k. How long until the sun gets burned down to the point where it cannot sustain life on earth anymore? The sun will last, at its current brightness for 9 billion more years.
How would you prove to them that earth orbits the sun? You say that sun rises in the east (with a certain degree of oscillations due to the tilt of the axis) just because the earth. Those are completely different things.
As i asked before, separate. The sun's spectrum is very complex, and indeed there are a lot of lines—both light and dark (emission and absorption)—amidst a sea of what looks to be continuous frequencies. It's consensus that the very similar apparent sizes of the moon and the sun as seen from earth is a coincidence (as already answered in this site). Similarly, how would you prove to them.
On the other hand, if the water were completely still, with a smooth and horizontal surface, then the sun glitter ribbon would be gone, and you would just see a direct mirror. 1) the sun seems brighter (more dazzling) if there is more scattering in the atmosphere. The sun does not rise, it is the horizon that goes down. The sun would actually look very small to us in the sky if there were no atmosphere (it's the same.