Monitoring CO2 plume migration and pressure buildup is critical for ensuring the safe and long-term containment of CO2 in geological formations during Geological CO2 Sequestration (GCS) processes. While reservoir simulators can consider full physics and predict high-fidelity flow dynamics in GCS, they often require much domain expertise to develop and high computational cost to predict. To alleviate these challenges, deep learning-based data-driven models have achieved significant progress in dynamics simulation in recent years, since they can achieve acceptable accuracy provided, they are trained on sufficient available simulation or field datasets. Unfortunately, the literature does not offer comprehensive benchmark solutions of different deep learning models, for complex GCS simulation cases. To bridge this necessary technical gap, we compare for a realistic but hypothetical storage reservoir the results from a well-accepted, robust commercial reservoir simulator with multiple deep neural network (DNN) models. The purpose is to simulate spatiotemporal patterns of CO2 plume migration and related pressure dynamics and further extend this to include dynamic geochemical reactions between fluid and minerals. Specifically, we evaluate seven DNN models including Fourier Neural Operator (FNO), UNet Enhanced Fourier Neural Operator (U-FNO), ResNet based Fourier Neural Operator (RU-FNO), UNet, ResNet, Attention UNet, and Generative Adversarial Networks (GANs). We first build a basic 2D radial reservoir model to simulate both CO2 injection and post-injection periods into a deep saline aquifer with proper boundary conditions. We further use the results to create a comprehensive simulation database with 2,000 cases, which cover a wide range of reservoirs and well parameters based on Latin Hypercube sampling approach. Among the seven models, the RUFNO model demonstrates robust performance, achieving an R2 score of 0.991 for saturation prediction and an R2 of 0.989 for pressure buildup prediction based on the blind testing dataset. The superior performance of RUFNO can be attributed to its combination of UNet-like architecture with the frequency-domain capabilities of Fourier Neural Operators that enhance their capability to predict complex reservoir behaviors. Given this superior performance, we further use RUFNO for geochemical reaction predictions, achieving R2 scores from 0.885 to 0.997 for different minerals. Further, in terms of computational efficiency, DNN models on average take 0.02 seconds/simulation run. This offers a speedup by orders of magnitude when compared to conventional reservoir simulation (these take on average 45 to 60 min/run). Therefore, DL models can deliver accurate and efficient predictions of both flow and geochemical dynamics in GCS and thus serve as a solid tool for GCS reservoir management for key parties in industry and government.