We perform extensive first-principles calculations for heterostructures composed of monolayer graphene and hexagonal boron nitride (hBN). Employing a symmetry-derived minimal tight-binding model, we extract orbital and spin-orbit coupling (SOC) parameters for graphene on hBN, as well as for hBN encapsulated graphene. Our calculations show that the parameters depend on the specific stacking configuration of graphene on hBN. We also perform an interlayer distance study for the different graphene/hBN stacks to find the corresponding lowest energy distances. For very large interlayer distances, one can recover the pristine graphene properties, as we find from the dependence of the parameters on the interlayer distance. Furthermore, we find that orbital and SOC parameters, especially the Rashba one, depend strongly on an applied transverse electric field, giving a rich playground for spin physics. Armed with the model parameters, we employ the Dyakonov-Perel formalism to calculate the spin relaxation in graphene/hBN heterostructures. We find spin lifetimes in the nanosecond range, in agreement with recent measurements. The spin relaxation anisotropy, being the ratio of out-of-plane to in-plane spin lifetimes, is found to be giant close to the charge neutrality point, decreasing with increasing doping, and being highly tunable by an external transverse electric field. This is in contrast to bilayer graphene in which an external field saturates the spin relaxation anisotropy.