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In this work, we present a lightweight, tightly-coupled deep depth network and visual-inertial odometry (VIO) system, which can provide accurate state estimates and dense depth maps of the immediate surroundings. Leveraging the proposed lightweight Conditional Variational Autoencoder (CVAE) for depth inference and encoding, we provide the network with previously marginalized sparse features from VIO to increase the accuracy of initial depth prediction and generalization capability. The compact encoded depth maps are then updated jointly with navigation states in a sliding window estimator in order to provide the dense local scene geometry. We additionally propose a novel method to obtain the CVAE's Jacobian which is shown to be more than an order of magnitude faster than previous works, and we additionally leverage First-Estimate Jacobian (FEJ) to avoid recalculation. As opposed to previous works relying on completely dense residuals, we propose to only provide sparse measurements to update the depth code and show through careful experimentation that our choice of sparse measurements and FEJs can still significantly improve the estimated depth maps. Our full system also exhibits state-of-the-art pose estimation accuracy, and we show that it can run in real-time with single-thread execution while utilizing GPU acceleration only for the network and code Jacobian.