When a feature dataset is selected in the Project Wizard, and a new feature dataset is specified as the output, only the first feature class in the selected feature dataset will be reprojected. The other feature classes will not be reprojected, nor added to the new feature dataset.
Unknown at this time.
Solution or Workaround
Run the Project Wizard again and select the remaining feature classes as inputs and the new feature dataset as the output.
Also, to avoid this in the future, one can create a new feature dataset with the correct projection first, then add all the individual feature classes from the source feature data set as inputs for the Project Wizard.