Pushing TOGA on the 737 with a single AP engaged automatically disengages it and you're into a hand flown go around. Equally, a single push of TOGA will not give you full beans GA thrust, but reduced thrust that will give you around 1000-2000ft fpm roc to avoid this kind of situation, and the FMA will reflect that, indicating GA not N1. You have to push TOGA twice to get full GA N1.
As for the NU stab trim, I thought it was part of the fail passive protections - ie if the autopilot chucks it at 100ft the aircraft will tend to pitch up and away from the ground?