Om 7.27, I don't understand the picture. I'm not sure how the wire can produce emf when the wire doesn't have an area of change. The answer says it's moving horizontally, but isn't the wire just moving without changing the area?

To think about this one, recognize there are free charges in the wire (sea of electrons, it's a conductor.)  Those charges are moving in a magnetic field.  Moving charges in a magnetic field experience a force, which will manifest itself in terms of an electric potential difference.