Do real and reactive energy exist?
Energy is just power integrated over time, so real and reactive energy exists or not just like real and reactive power, respectively.

As for power, real power exists and reactive power is a mathematical convenience to simplify expressing certain things.  By using the mental shortcut of imagining imaginary power, we can simplify calculations and explain real observed parameters more easily than without.

Imaginary power doesn't exist, but the effects of it projected back to real power are real.  Large scale power grid producers and consumers are often rated in both the instantaneous real and imaginary power they are producing or consuming.  However, the same real observable characteristics can be explained other ways.  Explaining them in terms of imaginary power is merely a mental and mathematical convenience.