1) Total eclipses are fairly localized events. They are a max of 166 miles wide. An eclipse doesn't plunge the world into darkness, just an area. 2) If Supergirl loses her powers during an eclipse, she would be useless at night when the Sun is on the other side of the Earth. However, it's important to note that the World Eater eclipse is not a natural phenomenon; it is generated by Kryptonian technology, not by the Earth's moon.
Katie McGrath slips into her native Irish accent at several points through this episode.