Why are gluons color charged but not photon? Could there be a charged EM force carriers like gluons or neutral color charge carrier like photon? Gluons have a color charge why don't photons have an electric charge like gluons?
 A: Being charged under a symmetry group means transforming in a non-trivial representation of the group.
The carrier boson of a force associated with a gauge symmetry transforms in the adjoint representation of the corresponding gauge group. For non-Abelian groups like the $\mathrm{SU}(3)$ of the strong force, the adjoint is non-trivial, and the bosons hence charged - their symmetry acts on them. For Abelian groups like the $\mathrm{U}(1)$ of electromagnetism, the adjoint is trivial and so acting with the symmetry on a photon does nothing, it is uncharged.
A: That is because when a photon is created from say an electron accelerating, it is the electric and magnetic fields that when moving in concert together in just the right way is considered an electromagnetic wave which is also the same thing as a photon. The electric and magnetic fields are just vectors and they do have a direction but they do not have charges assigned to them because these fields are neutral.
With gluons, the reason they are charged is because there are so many types of color charges and anticolor charges. These things can be combined in any possible way in order to create a gluon that has a net color charge. Technically speaking, the gluon has a color and an anticolor but these do not have to be the same which makes the gluon act like it has color.
With a photon, there are only two types of charges(positive or negative), and only one way they can combine resulting in a neutral photon. There is no off-balance that can occur which is why photons are neutral and gluons are charged.
A: Ultimately, the charges are a result of the Lie group symmetry of each field. Electrons are locally invariant under $U(1)$ symmetry, while quarks are locally invariant under $SU(3)$ symmetry. What these means is that certain transformations to quarks and leptons don’t change the behavior of the particles. Working this out mathematically gives the vector fields for gluons and photons. Pulling from representation theory, the number of generators for these groups/algebras (that is, quantities that can be added together to produce other elements of the group/algebra) are given by $N^2-1$, where $N$ refers to the group number ($SU(3)$ has $N=3$). This means gluons have 8 generators, while photons have 0 generators. These generators correspond to the charges/types of gluons/photons. Photons have no charged types (0 generators) while gluons have 8 charged types (8 generators).
A: To get back to the basics: physics is built on experiments. We start by assuming that the world exists and does whatever it does without giving any explanations. We can then do experiments or observations of the world. It is the experiments or observations that are "true", not our models.
Over time we have created models of the world. One of the models, the best one we have so far, is called the "standard model". This model has as example the colors of gluons. (Of course they are not colored, we simply call them colored as a convenience).
The "quality" of the model is how well it corresponds with our experiments and observations. You or anyone may design new models at any time. The "proof" of the model is when you do experiments or observations that show that your model is true, or perhaps false. In your example you could, as example, state that photons have electrical charge. Next  you design experiments to verify this and do the experiments and see if they support the model or not. We could see this kind of work when the standard model was tested for the existence of the so called Higgs boson: the theory predicted that it should exist, but it was only when it was observed in experiments that we could "prove" the validity of the model.
The "standard model" is an extremely good description of how the microscopic world behaves and has been verified to a very high precision. It is not without problems though, as there are unanswered questions. One of the questions is how to make it to work together with general relativity. In addition, it has a large number of "measured values" that we do not like, they have values but we want them to be derived from more fundamental things. There are a large number of alternative models in the works, you might have heard of as example string theory as one possible direction. Over time, I believe, we will create even better models of the world but when and how they will look is beyond me.
The standard model does not venture to describe "why" the world behaves the way it does. The model simply states that if you do this and that in an experiment you could expect a certain observation. So the standard model says there are certain aspects of gluons that can be described by the (totally made up) concept of color. This corresponds extremely well with experiments so it stays in the model. You may then turn around and say the the world behaves this way because our model says so, but that is not the reality. Instead, the world is what it is and our model should be a good description. Until we get a better model.
