Are the northern lights seen in the southern hemisphere simultaneously?

Apparently so. This phenomenon is called 'Auroral Conjugacy', and it was predicted as long ago as 1733. In fact, although no one could have carried out the experiment at that time, it was apparently common knowledge during the 19th century. Since charged particles are guided by magnetic fields, and since the field of the earth is a symmetric dipole, they felt secure back then in asserting that all auroras in the northern hemisphere would have simultaneous counterparts in the southern hemisphere. Because auroras are only easily seen at night, and occur at latitudes generally above 60 degrees, the difficulty in confirming this was that the appropriate antarctic regions are uninhabited, and are in nighttime when the north polar arctic regions are in daylight. It wasn't until 1968 that auroral conjugacy was confirmed by mounting cameras on aircraft and taking pictures.

Amazingly, not only do the aurora occur simultaneously on the same magnetic field line, but they are astonishingly similar in shape and intensity with the southern light a mirror image of the northern one. For more information about this, see Robert H. Eather's book "Majestic Lights" published in 1908 by the American geophysical Union, ISBN 0-87590-215-4.