This thesis deals with some cosmic aspects in the context of modified Gauss-Bonnet gravity. Firstly, we explore static spherically symmetric wormhole solutions in galactic halo region as well as using conformal Killing vectors technique. The effective energymomentum tensor leads to the violation of energy conditions while normal matter satisfies these conditions. We use Navarro-Frenk-White energy density profile to examine possible existence of traversable wormholes in galactic halo region. We find physically acceptable wormhole solutions threaded by normal matter for all values of r. We also investigate stability of the resulting wormhole solutions. For conformally symmetric traversable wormholes, it is found that all shape functions satisfy flaringout condition except phantom case with non-static conformal symmetry. Secondly, we study the dynamics of self-gravitating objects for spherical and axial systems. We construct structure scalars through orthogonal splitting of the Riemann tensor and deduce a complete set of equations governing the evolution of dissipative anisotropic fluid in terms of these scalars. In spherically symmetric system, we investigate some particular fluid models according to various dynamical conditions and find that our results are consistent with general relativity for constant f(G) model. Any other choice of the model leads to irregular distribution of dark energy and deviates from general relativity. We also explore different causes of density inhomogeneity which turns out to be a necessary condition in the presence of dark sources. In axially symmetric system with shear, it is found that dark sources affect thermodynamics of the system, evolution of kinematical quantities as well as density inhomogeneity. For the shear-free case, we study both non-geodesic as well as geodesicfluids with and without dissipation. The non-geodesic (non-dissipative) fluid gives inhomogeneous expansion while geodesic fluid leads the system either to vorticityfree or expansion-free. The vorticity-free non-dissipative geodesic fluid reduces the axial system to FRW model with homogeneous distribution of dark sources while expansion-free geodesic fluid does not exist even in the presence of dark sources.