The increasing availability of Internet of Things (IoT) applications has led to the development of new technologies. Specifically, the deployment of edge servers close to IoT devices has strengthened the edge computing paradigm. With the collaboration of Mobile Edge Computing (MEC) and cloud computing, delay-sensitive and computation-intensive tasks can be offloaded to the edge/cloud servers to improve system performance in terms of the delay and energy consumption of IoT devices. However, there is a need to schedule the computation tasks for an efficient management. More importantly, the task scheduling strategy can face data tampering attacks to deliberately modify, destroy or manipulate the decisions. To solve the above problems, in this paper, we newly propose to integrate digital twin and blockchain into the edge networks. However, it is unclear (i) how energy and delay-aware computation should be combined, and (ii) which mining computations should be executed for a secure task scheduling. The state-of-the-art focuses on task scheduling and blockchain mining, separately. Therefore, we propose a novel blockchain-based digital twin-edge network architecture where our proposed algorithm solves these two challenges at the same execution. We design a three-layer system architecture, composed of physical entity layer, digital twin edge layer and blockchain layer. In the physical entity layer, we formulate an energy and delay-aware task scheduling problem. In the digital twin edge layer, we propose a novel Proof of Evaluation (PoE)-based secure energy and delay-aware task scheduling algorithm where optimization is executed by the genetic algorithm implementation of Warehouse Location Problem (WLP). In the blockchain layer, the best-found solutions are shared with the topology in a blockchain. Here, each block includes the hash of the previous block, a genetic algorithm-based solution, nonce value, and a hash of whole blocks in the blockchain. Thus, we aim to execute the computation tasks with an acceptable delay in an energy-efficient manner and prevent data tampering attacks against the optimal computation decisions. We validate the outcomes of our PoE-based secure digital twin-edge network model with extensive evaluations. Since the proposed model distributes the task not only to the local device but also to the MEC and cloud server for delay awareness, it reduces the delay but consumes more energy. Nevertheless, the additional energy consumption can be neglected against the delay reduction. The proposed scheme is also more scalable to compare with the conventional solution. The numerical results clearly show that the proposed model provides energy and delay awareness, maintaining both data integrity and trustworthiness at the same execution of algorithm.