Yes, fuel economy can improve with a diesel remap. My 530d went from 6.8 to 6.2 L/100km on highway after stage 1. The engine is more efficient when it doesnt have to work as hard.
Yes, diesel tunes can improve fuel economy. My 530d went from 6.5L/100km to about 6.0L/100km on highway after stage 1. The reason is the engine makes more torque at lower RPM, so you can cruise at lower revs. In the city the difference is smaller because you tend to use the extra power. Overall I save about 0.3-0.5L/100km in mixed driving.
The B57 in the 530d is one of the best diesel engines ever made. Stage 1 gives you about 310-320hp and 650nm. The fuel economy improvement is real but depends on your driving style. If you drive the same way as before, youll use less fuel. If you enjoy the extra power (and you will), consumption stays about the same.