What the black body radiation stand for?

i'm having hard time to figure out what is happening in a black body in thermodynamic equilibrium. Let me tell what i know so far: by definition the black body emits and absorb all radiation from and to the environment continuously in a energetic balanced way, i learned that we can model this situation with standing waves: on the surface of the black body the electric field is constantly zero.

1) I visualize a part of the energy as stored in the electromagnetic field inside the black body. Is this energy all the energy of the black body? I think no because if we assume a black body full of gas for example, some energy will be stored in the kynetic and potential energy of the particles.

2) In the classical argument of the Rayleigh-Jeans in HyperPhysics by mean of the equipartition principle each mode will have the same average energy $$KT$$.Do the modes count as degree of freedom? Why? I think all degree of freedom of the system are only the translation, rotation and vibration of the molecules of the gas (I really don't get this mix of kynetic theory of gas and electromagnetism). If i assume as example of black body a cavity this get more sense to me but still i don't get the whole picture.

3) Are the modes of the standing waves the molecules vibrational modes?If yes, why? How can i visualize that? This would means that all the energy of the black body is stored in thermal energy (translational,rotational,vibrational) and the standing waves are just a way to represent this energy (though i wouldn't get how this account for translational and rotational).

I really don't get the whole picture, the mechanism behind this phenomena.

by definition the black body emits and absorb all radiation from and to the environment continuously in a energetic balanced way

By definition, black body absorbs all radiation coming its way. It is not necessarily emitting the same amount of energy. So there is not necessarily an energetic balance; black body can radiate more energy than it accepts, and be cooling down.

i learned that we can model this situation with standing waves: on the surface of the black body the electric field is constantly zero.

No, there is no such condition for a general black body. Electric field is zero on the surface of perfectly conducting walls of a cavity. This is an hypothetical idealized system that is one example of how black body can be made and analyzed.

1) I visualize a part of the energy as stored in the electromagnetic field inside the black body. Is this energy all the energy of the black body? I think no because if we assume a black body full of gas for example, some energy will be stored in the kynetic and potential energy of the particles.

Yes, EM energy is part of total energy of real bodies. It is total energy only when the black body is the artificial one - equilibrium radiation in a perfectly reflecting cavity.

2) In the classical argument of the Rayleigh-Jeans in HyperPhysics by mean of the equipartition principle each mode will have the same average energy $$KT$$.Do the modes count as degree of freedom? Why?

Mode is an independent way of oscillation. Assuming EM energy is given by the Poynting energy of the region in the cavity, which is quadratic function of EM fields, we can re-express energy in terms of contributions due to harmonic oscillation modes (stronger oscillation means stronger contribution to energy). Since each such term is a quadratic function of canonical variables (Fourier coefficients and its time derivatives), people thought "this is similar to harmonic oscillator in mechanics, so let's apply the equipartition theorem". So it is a formal analogy, and in this analogy each mode of harmonic oscillation has energy $$kT$$.

I think all degree of freedom of the system are only the translation, rotation and vibration of the molecules of the gas (I really don't get this mix of kynetic theory of gas and electromagnetism). If i assume as example of black body a cavity this get more sense to me but still i don't get the whole picture.

Yes, if the black body contains matter (molecules, atoms), then the Rayleigh-Jeans model is deficient. It only deals with EM energy in empty space, it does not account for matter. So its naive application to material black body is incorrect.

3) Are the modes of the standing waves the molecules vibrational modes?If yes, why? How can i visualize that? This would means that all the energy of the black body is stored in thermal energy (translational,rotational,vibrational) and the standing waves are just a way to represent this energy (though i wouldn't get how this account for translational and rotational).

No, the modes in the Rayleigh-Jeans model are modes of EM radiation in an empty space, or in a perfectly reflecting cavity. Modes of oscillation of molecules are different modes, and they are not accounted for in that model.