An alternative view: Electric charge is the relative direction of imaginary lines of force in an electric field. Electric field is circular in nature. As charge is a relative entity, type of charge depends on reference used and all electric fields have both positive and negative electric charges. Electric charge is a functional entity.
Mas is another functional entity that shows mathematical relation between external force on a body and body's acceleration in the direction of force. Mass is often considered to represent equivalent of quantity of three-dimensional matter in a body.
With sufficient deliberations, any type of relation can be established between functional entities.
Electrically charged particles cannot be massless because of their self-energy. The field that is constituted by the charge contains some energy and contributes to the mass of the charged particle because of the equivalence of mass and energy (“E=mc2”). Thus, the mass of an electrically charged particle cannot be exactly zero. I.e., there is no electrical charge without mass.
For the color-charged gluons the situation is different due to their much more complicated self-interacting terms. They underlie a manifest gauge symmetry, which is not spontaneously broken (as it is in the electroweak sector). This “forbids” massive gluons.
Thus, to my current knowledge, I have no reason to doubt the masslessness of gluons, even if every electrically charged particle must have a (rest) mass.
If there is any reason for gluons to have mass (e.g. by experiment), there would be a need to revise QCD. (However, as already stated, currently I do not know any evidence for that.)
(However, I do not guarantee for the correctness of these explanations, since I am not the author.)
I am not sure whether I really understand your question about the difference between mass and energy. At least there is no difference, because mass and energy are equivalent.
Furthermore, I do not see the very difference between the definitions of mass and energy within classical physics (including relativity) and quantum physics. However, there is - in fact - a difference between classical and quantum physics concerning the concept of self-energy.
The main difference between mass and energy is 'Inertia'. Of course Now I don't want to get in to that discussion. Thank you. I will go through your explanations.