According to the advances in quantum computing and distributed learning, quantum federated learning (QFL) has recently become an emerging field of study. In QFL, each quantum computer or device locally trains its quantum neural network (QNN) with trainable gates, and communicates only these gate parameters over classical channels, without costly quantum communications. To successfully opeate QFL under various and dynamic channel conditions in Internet of Things (IoT) environments, this article develops a novel depth-controllable architecture of entangled slimmable QNNs (eSQNNs), and thus, proposes an entangled slimmable QFL (eSQFL) that communicates the superposition-coded parameters of eSQNNs. Even though the proposed eSQNN-based eSQFL is superior, training the depth-controllable eSQNN architecture is challenging due to high-entanglement entropy and interdepth interference. Therefore, the proposed method in this article mitigates the interference using entanglement controlled universal (CU) gates and an inplace fidelity distillation (IPFD) regularizer penalizing interdepth quantum state differences, respectively. Furthermore, the proposed method optimizes the superposition coding power allocation by deriving and minimizing the convergence bound of eSQFL. The novelty of this work is evaluated via extensive simulations in terms of prediction accuracy, fidelity, and entropy compared to Vanilla QFL as well as under different channel conditions and various data distributions.