  Van der Waals (vdW) heterostructures consisting of Bernal bilayer graphene
(BLG) and hexagonal boron nitride (hBN) are investigated. By performing
first-principles calculations we capture the essential BLG band structure
features for several stacking and encapsulation scenarios. A low-energy model
Hamiltonian, comprising orbital and spin-orbit coupling (SOC) terms, is
employed to reproduce the hBN-modified BLG dispersion, spin splittings, and
spin expectation values. Most important, the hBN layers open an orbital gap in
the BLG spectrum, which can range from zero to tens of meV, depending on the
precise stacking arrangement of the individual atoms. Therefore, large local
band gap variations may arise in experimentally relevant moir\'{e} structures.
Moreover, the SOC parameters are small (few to tens of $\mu$eV), just as in
bare BLG, but are markedly proximity modified by the hBN layers. Especially
when BLG is encapsulated by monolayers of hBN, such that inversion symmetry is
restored, the orbital gap and spin splittings of the bands vanish. In addition,
we show that a transverse electric field mainly modifies the potential
difference between the graphene layers, which perfectly correlates with the
orbital gap for fields up to about 1~V/nm. Moreover, the layer-resolved Rashba
couplings are tunable by $\sim 5~\mu$eV per V/nm. Finally, by investigating
twisted BLG/hBN structures, with twist angles between 6$^{\circ}$ --
20$^{\circ}$, we find that the global band gap increases linearly with the
twist angle. The extrapolated $0^{\circ}$ band gap is about 23~meV and results
roughly from the average of the stacking-dependent local band gaps. Our
investigations give new insights into proximity spin physics of hBN/BLG
heterostructures, which should be useful for interpreting experiments on
extended as well as confined (quantum dot) systems.
