The covariant density functional (CDF) theory with a few number of parameters has been successfully employed to describe ground-state and excited-states of nuclei over the entire nuclear landscape for A > 12. It describes finite nuclear systems with a universal hadronic Lagrangian, which is solved considering the relativistic-Hartree-Fock-Bologuibov approach (RHFB). This model is also employed for the study of compact stars, since it can be extended to high densities where special relativity cannot be ignore. This model can also be extended to include the contribution of hyperons and as well as other exotic particles. In this work, the description and some predictions based on RHFB approach for nuclei under extreme conditions of mass, isospin and temperature are presented.In the first part, we explore the occurrence of spherical shell closures for superheavy nuclei, where shell closures are characterized in terms of two-nucleon gaps. The results depend slightly on the effective Lagrangians used, but the magic numbers beyond ^{208}Pb are generally predicted to be Z = 120 and 138 for protons, and N = 172, 184, 228, and 258 for neutrons. Shell effects are sensitive to various terms of the mean-field, such as the spin-orbit coupling, the scalar and the effective masses, as well as the Lorentz-tensor interaction. These terms have different weights in the effective Lagrangians employed, explaining the (relatively small) variations in the predictions. Employing the most advanced RHFB model, we founded that the nuclide ^{304}120 is favored as being the next spherical doubly-magic nucleus beyond ^{208}Pb.In the second part, we investigate the formation of new shell gaps in intermediate mass neutron-rich nuclei, and analyze the role of the Lorentz pseudo-vector and tensor interactions. Based on the Foldy-Wouthuysen transformation, we discuss in detail the role played by the different terms of the Lorentz pseudo-vector and tensor interactions in the appearance of the N=16, 32 and 34 shell gaps. The nuclei ^{24}O, ^{48}Si and ^{52,54}Ca are predicted with a large shell gap and zero (^{24}O, ^{52}Ca) or almost zero (^{48}Si, ^{54}Ca) pairing gap, making them candidates for new magic numbers in neutron rich nuclei. We find that the Lorentz pseudo-vector and tensor interactions induce very specific evolutions of single-particle energies, which could clearly sign their presence and reveal the need for relativistic approaches with exchange interactions.In the last part, we study the phase transitions and thermal excitations of both stable and weakly-bound nuclei. The predictions of various relativistic Lagrangians and different pairing interactions are discussed. The critical temperature of the pairing transition is found to depend linearly on the zero-temperature pairing gap, and this dependence is similar for a zero-range or a finite-range pairing interaction. The present calculations show interesting features of the pairing correlations at finite temperature, such as the pairing persistence and pairing re-entrance phenomena. Also, we analyze the thermal response of some nuclei.In conclusion, the work presented in this thesis shown interesting and new results for three of the most important questions in nuclear physics: the quest for a new island of stability in the superheavy region, the appearance of new magic numbers in exotic nuclei, and the response of finite-systems to thermal excitations.