Optical lattice clocks have reached uncertainties in 10^{-18} regime, well surpassing the primary microwave frequency standard. Such performance levels have allowed for applications from geodesy to fundamental physics. The performance of state of the art optical lattice clocks are strongly influenced by black body radiation (BBR) induced frequency shifts. Magnesium is one of the optical lattice clock candidate elements with very low sensitivity to BBR, which makes it an interesting candidate as an optical frequency reference.Optical lattice clocks rely on high-Q optical transitions, where Doppler and recoil shifts are suppressed by trapping the atoms in Lamb-Dicke regime. For Magnesium, due to its low atomic mass, the tunneling induced line-broadening is significantly large. This has been a bottleneck in reducing the instability of Magnesium lattice clock. However the large tunneling rate for Magnesium atoms in the optical lattice also allows us to study these lattice effects using optical spectroscopy.Lattice AC Stark shift is one of the important contributions to the uncertainty budget for an optical lattice clock. To achieve clock uncertainties in 10^{-18} regime, even the contributions from multipolar polarizabilities and hyperpolarizability becomes significant. Therefore, operational magic frequencies have been identified in Strontium and Ytterbium lattice clocks, where the light shift dependence on intensity is zero to the lowest order.In this thesis, an extensive model has been developed to understand the influence of tunneling in a one dimensional optical lattice on the clock transition lineshape. This model is used to simulate the spectroscopy results previously observed in our experiment, which show strong lineshape asymmetry as lattice wavelength is detuned from the magic condition. The strong influence of transverse states in generating these asymmetries was highlighted by numerical simulations.To improve the performance of our Magnesium lattice clock from the last frequency measurements, lattice system upgrades were carried out within the scope of this thesis. This allowed to suppress the tunneling induced line-broadening to sub-Hz regime for the first time for magnesium, and to resolve the 1S0-3P0 clock transition with a linewidth of 7(3) Hz. The high line-Q thus obtained of 9(3) x 10^{13} helped reduce the clock instability in self-comparison measurement to 7.2^{+7.7}_{-1.8} x 10^{-17} in 3000 seconds of averaging time.The improved clock instability also helped estimate various systematic shifts with much improved uncertainties. The probe AC Stark shift and Zeeman shift uncertainties were reduced to the mid-10^{-17} regime, while cold collision shift was characterized with uncertainty of 1.4 x 10^{-16}. With an aim to similarly reduce lattice AC Stark shift uncertainty, influence of higher order shifts was characterized for Magnesium for the first time. The hyperpolarizability coefficient was estimated to be 197(53) micro Hz/(kWcm^{-2})^2. These measurements show that the lattice shift can be characterized with an uncertainty of 6.5 x 10^{-16}, paving way for a future frequency measurement with more than an order of magnitude lower uncertainty.