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Multi-objective resource allocations are studied for a multi-level edge-caching enabled fog-radio access network. Notably, joint maximization of the energy-efficiency (EE) and spectrum-efficiency (EE) and interference management for the content distribution phase are investigated. In the envisioned system, the popular contents are cached at both the fog access point (F-AP) and fog user equipments (F-UEs), and the content-requesting devices are grouped into multiple device-clusters based on their locations. Using a rate-splitting with common message decoding based transmission strategy, each device-cluster is simultaneously served by a suitably selected F-UE and the F-AP over the same radio resource blocks. To maximize system EE and SE of the content distribution phase jointly, a multi-objective optimization problem (MOOP) is formulated. By using the $\epsilon$-constraint method, the proposed MOOP is converted to an EE-SE trade-off optimization problem. Leveraging iterative function evaluation based power control and generalized 3D-resource matching, a novel algorithm is devised to obtain near-optimal Pareto front for the proposed MOOP. By inspecting solutions over the Pareto optimal front, a suitable operating EE-SE pair and the corresponding resource allocation variables are obtained. A low-complexity algorithm to obtain a suitable EE-SE trade-off is also devised. The conducted simulations confirm that the proposed algorithms achieve substantial improvement of system EE and SE over the baseline schemes.