Regarding first part of the problem (low-load motor fed by VFD):
If the motor operates in constant torque region, VFD keeps the flux constant unless special precaution is taken in the drive. So this means at low load the motor flux which is an indicator of the power factor will be the same as rated power. On the other hand active power drops due to the low load operation. As a result power factor will drop for low load operation. However, if some efficiency improvement scheme is implemented in the drive, then power factor should remain constant as you lower the load.
If the motor operates in the constant power region, flux will be reduced with incresing speed to keep V/f constant. So depending on which speed you are operating, power factor may remain constant, incerase or decrease. However if you make sure proportional flux reduction with lowering load, than you will certainly have constant or increasing power factor.