We present a systematic DFT-GGA theoretical characterization of the RE₄Fe₂As₂Te_1−xO_4−yF_y family of compounds (conventionally called 42214) as a function of a set of key tuning parameters: rare earth (RE = Pr, Sm, and Gd), Te content, oxygen → fluorine substitutional doping, and external pressure. We focus our discussion on the effect of these parameters on magnetic stability and on the nonmagnetic electronic structure, as most relevant aspects related to the occurrence of superconductivity. To uncover the complexity of the 42214 crystal structure, the electronic analysis is based on an unfolding procedure that allows us to observe the behavior of the hole and electron pockets of the Fermi surface and of the nesting function. We complete the present study with the characterization of a related hypothetical compound having Se substituting for Te. Our results show that this peculiar compound family offers very good opportunities to properly harness material properties; based on our results, we infer that suitably tuning a variety of parameters, as those examined here, improved superconducting properties could be achieved.