Device-edge co-inference opens up new possibilities for resource-constrained wireless devices (WDs) to execute deep neural network (DNN)-based applications with heavy computation workloads. In particular, the WD executes the first few layers of the DNN and sends the intermediate features to the edge server that processes the remaining layers of the DNN. By adapting the model splitting decision, there exists a tradeoff between local computation cost and communication overhead. In practice, the DNN model is re-trained and updated periodically at the edge server. Once the DNN parameters are regenerated, part of the updated model must be placed at the WD to facilitate on-device inference. In this paper, we study the joint optimization of the model placement and online model splitting decisions to minimize the energy-and-time cost of device-edge co-inference in presence of wireless channel fading. The problem is challenging because the model placement and model splitting decisions are strongly coupled, while involving two different time scales. We first tackle online model splitting by formulating an optimal stopping problem, where the finite horizon of the problem is determined by the model placement decision. In addition to deriving the optimal model splitting rule based on backward induction, we further investigate a simple one-stage look-ahead rule, for which we are able to obtain analytical expressions of the model splitting decision. The analysis is useful for us to efficiently optimize the model placement decision in a larger time scale. In particular, we obtain a closed-form model placement solution for the fully-connected multilayer perceptron with equal neurons. Simulation results validate the superior performance of the joint optimal model placement and splitting with various DNN structures.