We report scanning tunneling spectroscopic (STS) studies of the low-energy quasiparticle excitations of cuprate superconductors as a function of magnetic field and doping level. Our studies suggest that the origin of the pseudogap (PG) is associated with competing orders (COs), and that the occurrence (absence) of PG above the superconducting (SC) transition Tc is associated with a CO energy ΔCO larger (smaller) than the SC gap ΔSC. Moreover, the spatial homogeneity of ΔSC and ΔCO depends on the type of disorder in different cuprates: For optimally and under-doped YBa2Cu3O7−δ (Y-123), we find that ΔSC<ΔCO and that both ΔSC and ΔCO exhibit long-range spatial homogeneity, in contrast to the highly inhomogeneous STS in Bi2Sr2CaCu2O8+x (Bi-2212). We attribute this contrast to the stoichiometric cations and ordered apical oxygen in Y-123, which differs from the non-stoichiometric Bi-to-Sr ratio in Bi-2212 with disordered Sr and apical oxygen in the SrO planes. For Ca-doped Y-123, the substitution of Y by Ca contributes to excess holes and disorder in the CuO2 planes, giving rise to increasing inhomogeneity, decreasing ΔSC and ΔCO, and a suppressed vortex-solid phase. For electron-type cuprate Sr0.9La0.1CuO2 (La-112), the homogeneous ΔSC and ΔCO distributions may be attributed to stoichiometric cations and the absence of apical oxygen, with ΔCO<ΔSC revealed only inside the vortex cores. Finally, the vortex-core radius (ξhalo) in electron-type cuprates is comparable to the SC coherence length ξSC, whereas ξhalo∼10ξSC in hole-type cuprates, suggesting that ξhalo may be correlated with the CO strength. The vortex-state irreversibility line in the magnetic field versus temperature phase diagram also reveals doping dependence, indicating the relevance of competing orders to vortex pinning.