An EMF is generated whenever a conductor moves relative to a magnetic field so that the conductor is cutting across the magnetic field lines. The EMF generated is the cross product of the magnetic field and the motion of the conductor (I may have the sign flipped, don't remember off the top of my head).
To answer your rotating rod question: yes EMF is produced inside the rod. However, note that this EMF will not be end to end of the rod, but from each end to its center.
If you attached brushes to the ends of the rod such that they touched the inside of a fixed cylinder, then the whole apparatus would be a generator between this cylinder and the center of the rod. If you connected this generator to a load, current would now flow, which now produces a force on the rod, which would oppose its motion.