This paper introduces two neural network-based methods for simultaneously computing lower and upper bounds of American-style option prices, along with hedging strategies. The first method uses a series of neural networks, one per exercise time, to approximate continuation values and martingale increments via regression. The second method employs a single global network that takes time as an additional input, requiring an iterative training-updating procedure to refine stopping strategies. Both methods avoid nested Monte Carlo simulations, significantly reducing computational complexity. The dual formulation of the option price is leveraged to derive upper bounds, and the martingale representation directly yields hedging strategies without differentiation. Numerical experiments on vanilla American put options and max-call options in dimensions up to 10 demonstrate that the methods produce tight bounds and effective hedges. Algorithm variations, such as warm-start training and fresh data generation, further improve efficiency and memory usage. The hedging strategies can also serve as control variates to reduce variance in lower bound estimates.