Motivated by the drawbacks of cloud-based federated learning (FL), cooperative federated edge learning (CFEL) has been proposed to improve efficiency for FL over mobile edge networks, where multiple edge servers collaboratively coordinate the distributed model training across a large number of edge devices. However, CFEL faces critical challenges arising from dynamic and heterogeneous device properties, which slow down the convergence and increase resource consumption. This paper proposes a heterogeneity-aware CFEL scheme called \textit{Heterogeneity-Aware Cooperative Edge-based Federated Averaging} (HCEF) that aims to maximize the model accuracy while minimizing the training time and energy consumption via adaptive computation and communication compression in CFEL. By theoretically analyzing how local update frequency and gradient compression affect the convergence error bound in CFEL, we develop an efficient online control algorithm for HCEF to dynamically determine local update frequencies and compression ratios for heterogeneous devices. Experimental results show that compared with prior schemes, the proposed HCEF scheme can maintain higher model accuracy while reducing training latency and improving energy efficiency simultaneously.