What are chromoelectric and chromomagnetic fields? Are they the normal electric and magnetic fields from Maxwell fields? Or are they just the corresponding components from $G_{\mu\nu}^a$ (the gluon fields), say chromoelectric fields are simply $G_{0i}^a$ ($i=1,2,3$), and chromomagnetic fields are simply $G_{ij}^a$ ($i,j=1,2,3$).
 A: No they are not normal electric and magnetic fields.
Quantum chromodynamics is a quantum field theory, that describes the strong interaction of quarks and gluons employing the concept of color charge.
Chromomagnetism is an interaction between quarks of different color that has some similarities to magnetism.
Chromoelectricity is an interaction between quarks of different color that has some similarities to electricity.
Now chromoelectric and chromomagnetic fields are the fields generated by gluons, you are correct.
You are correct that they show some similarities to the normal EM fields, but in the case of EM fields, there are two types of charges and they are attractive or repulsive, while with ChromoEM fields, there are three types of charges and they are attractive.
Please see here:

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*M. Cardoso, N. Cardoso, P. Bicudo, Chromoelectric and chromomagnetic fields for the static gluon-quark-antiquark system, PoS LAT2009:233,2009, arXiv:0910.0133.

The strong force is relativistic. You are correct, that you can define chromoEM fields from the strong field strength tensor, this is similar to EM fields, and you can use Lorentz transformations so that the moving ChromoE field generates a ChromoM field.
A: The classical field theory of chromodynamics is a classical field theory based upon SU(3) and is modelled in analogy to electrodynamics which is modelled upon U(1).
The electromagnetic field tensor incorporates both the electric and magnetic field strengths; likewise, I suspect the corresponding field tensor in chromodynamics incorporates certain physically measurable fields in the theory of quarks and gluons but I'm not sure about this.
The electromagnetic field strength is the exterior derivative of the electromagnetic potential; and it's a good question which one is more fundamental. Classically, it was thought that the field strengths, being measurable, were the fundamental quantities and the potential an intriguing mathematical convenience.
However, quantum effects in the Aharonov-Bohm Effect plausibly show that the potential appears to be the more fundamental quantity. Likewise one may suppose the same in Chromodynamics.
