  We investigate in detail the $\nu=+1$ displacement-field-tuned interacting
phase diagram of $L=3,4,5,6,7$ layer rhombohedral graphene aligned to hBN
(R$L$G/hBN). Our calculations account for the 3D nature of the Coulomb
interaction, the inequivalent stacking orientations $\xi=0,1$, the effects of
the filled valence bands, and the choice of `interaction scheme' for specifying
the many-body Hamiltonian. We show that the latter has a dramatic impact on the
Hartree-Fock phase boundaries and the properties of the phases, including for
pentalayers (R5G/hBN) with large displacement field $D$ where recent
experiments observed a Chern insulator at $\nu=+1$ and fractional Chern
insulators for $\nu<1$. In this large $D$ regime, the low-energy conduction
bands are polarized away from the aligned hBN layer, and are hence
well-described by the folded bands of moir\'eless rhombohedral graphene at the
non-interacting level. Despite this, the filled valence bands develop
moir\'e-periodic charge density variations which can generate an effective
moir\'e potential, thereby explicitly breaking the approximate continuous
translation symmetry in the conduction bands, and leading to contrasting
electronic topology in the ground state for the two stacking arrangements.
Within time-dependent Hartree-Fock theory, we further characterize the strength
of the moir\'e pinning potential in the Chern insulator phase by computing the
low-energy $\mathbf{q}=0$ collective mode spectrum, where we identify competing
gapped pseudophonon and valley magnon excitations. Our results emphasize the
importance of careful examination of both the single-particle and interaction
model for a proper understanding of the correlated phases in R$L$G/hBN.
