What is energy? 

The sun is not shooting out light/em wave/photon/energy to every direction, it only shares its vibrational em force with matters surround. the energy/force strength is proportional to sun's temperature and inversely proportional to distance square. 

The sun's energy is from star formation, not from ongoing nuclear reaction.

The sun will never cool off a bit, without other huge mass come closer.

It won't happen soon, the closest star is 4 light years away.

If Earth temperature suddenly becomes as high as the Sun, what would happen?

All nuclear fuel will be instantly ignite and release energy, after that no more nuclear reaction at all.

That is the Sun's fate also.