Using density functional theory, we investigate the interplay among the sequence of bilayers composed of an Fe and a Rh layer on the Re(0001), their stacking order and their magnetic properties. We find that bilayers in which both layers are in fcc stacking are energetically very unfavorable, while all other combinations of hcp and fcc stacking are energetically relatively close. The magnetic interactions are evaluated by mapping the DFT energies onto an atomistic spin model, which contains the Heisenberg exchange, the Dzyaloshinskii-Moriya interaction (DMI), the magnetocrystalline anisotropy energy, and higher-order exchange interactions. We find that the stacking sequence of the bilayer significantly modifies the magnetic interactions. For bilayers in which Fe is the topmost layer, the nearest-neighbor exchange interaction is ferromagnetic, but varies in strength by a factor of up to three for different stacking sequences. The DMI even changes up to a factor of four. As a result, we find a DMI driven cycloidal spin spiral ground state with a period of 11 nm for hcp-Fe/hcp-Rh/Re(0001). For fcc-Fe/hcp-Rh/Re(0001) and hcp-Fe/fcc-Rh/Re(0001), we obtain a ferromagnetic ground state. The spin spiral energy dispersion of hcp-Fe/hcp-Rh/Re(0001) including spin-orbit coupling suggests that isolated skyrmions can be stabilized in the field-polarized ferromagnetic background at external magnetic fields. If the Fe layer is sandwiched between the Rh overlayer and the Re(0001) substrate, there is a competition between the ferromagnetic coupling preferred by the Rh-Fe hybridization and the antiferromagnetic coupling induced by the Fe-Re hybridization. Due to the Fe/Re interface, the DMI can become very large. For fcc-Rh/hcp-Fe/Re (0001), we obtain a cycloidal spin spiral with a period of 1.7 nm, which is induced by frustration of exchange interactions and further stabilized by the DMI. For hcp-Rh/hcp-Fe/Re(0001), we find a DMI driven cycloidal spin spiral with a period of 4 nm and locally nearly antiparallel magnetic moments due to antiferromagnetic nearest-neighbor exchange. The higher-order exchange constants can be significant in the considered films, however, they do not stabilize multi-Q states.