If light is a kind of energy and energy is massless so how does light exists

And how can a particle/wave can be emitted by an electron billions of times and the electron's state and mass doesn't change.

When an electron goes from a high state to a lower state and emits a photons, it does lose a tiny bit of mass. The reverse is true. When an atom absorbs a photon, and so an electron goes from a low energy state to a high energy state, the atom gains energy and so mass.

The mass difference is very tiny and difficult to measure, but since larger energy differences (nuclear ones) can produce measurable mass differences, we can pretty certain that the small mass differences are there.

Light isn't really a kind of energy. It has kinetic energy. And we can talk about light energy, the energy that light transmits. But it also has momentum and we don't really say that "light is a kind of momentum". It also has angular momentum.

I like that you are trying to work things out, but you are pretty mixed up. I recommend Richard Feynman's book "QED: A Strange Theory of Light and Matter" as a place to start. It was written for a nontechnical audience and is a great introduction to ideas of quantum mechanics.

An electron's state DOES change when it emits a photon. And the mass of the atom alters also by the amount of energy ( E = mc^2) emitted in the photon.

If there is not an ongoing source of energy then the light diminishes.
Try any of the "glow in the dark" materials. They are initially bright and gradually become dimmer.
This is because each electron can only change its state once unless some more energy is added.

But if we add energy in a range of ways this is no longer true.
Heat a metal to white heat. The heat raises the energy levels of the electron and adds energy and mass to it. Subsequently the electron changes its orbit and emits a photon.
Of course this also cools the metal slightly. A fact that can be seen if we remove the source of heat.