To meet the diverse needs of users, the rapid advancement of cloud-edge-device collaboration has become a standard practice. However, this complex environment, particularly in untrusted (non-collaborative) scenarios, presents numerous security challenges. Authentication acts as the first line of defense and is fundamental to addressing these issues. Although many authentication and key agreement schemes exist, they often face limitations, such as being tailored to overly specific scenarios where devices authenticate solely with either the edge or the cloud, or being unsuitable for resource-constrained devices. To address these challenges, we propose an adaptive and efficient authentication and key agreement scheme (AEAKA) for Cloud-Edge-Device IoT environments. This scheme is highly adaptive and scalable, capable of automatically and dynamically initiating different authentication methods based on device requirements. Additionally, it employs an edge-assisted authentication approach to reduce the load on third-party trust authorities. Furthermore, we introduce a hash-based algorithm for the authentication protocol, ensuring a lightweight method suitable for a wide range of resource-constrained devices while maintaining security. AEAKA ensures that entities use associated authentication credentials, enhancing the privacy of the authentication process. Security proofs and performance analyses demonstrate that AEAKA outperforms other methods in terms of security and authentication efficiency.