While the musculoskeletal humanoid has various biomimetic benefits, the modeling of its complex structure is difficult, and many learning-based systems have been developed so far. There are various methods, such as control methods using acquired relationships between joints and muscles represented by a data table or neural network, and state estimation methods using Extended Kalman Filter or table search. In this study, we construct a Musculoskeletal AutoEncoder representing the relationship among joint angles, muscle tensions, and muscle lengths, and propose a unified method of state estimation, control, and simulation of musculoskeletal humanoids using it. By updating the Musculoskeletal AutoEncoder online using the actual robot sensor information, we can continuously conduct more accurate state estimation, control, and simulation than before the online learning. We conducted several experiments using the musculoskeletal humanoid Musashi, and verified the effectiveness of this study.