You could calculate this by finding the position of the Moon on the celestial sphere, and then figuring out what point on the Earth’s surface this corrseponds to. This ephemeris table gives you the information about the right ascension & declination of the Moon. The declination of the Moon (at the bottom) will be equal to the latitude of the spot on Earth’s surface you’re looking for (positive = north, negative = south.) To find the longitude of the spot you’re looking for, you would subtract the Sun’s right ascension from the Moon’s right ascension; positive results would then be East longitude, and negative would be West longitude.
For the time & date you mentioned, I get 5.867°E, 20.64°S, which is in the South Atlantic, about 800 km off the coast of Namibia.
No, because the Earth rotates and revolves about the Sun. If the Earth wasn’t rotating or revolving, though, it would be pretty close to a great circle. Great Antibob notes that the orbit itself is an ellipse; however, this ellipse lies in a plane, and the intersection of this orbital plane with the Earth is what the “path of the cord” would be. Assuming a perfectly spherical Earth, this would be a great circle.
Because of the motion of the Earth, it takes a while to complete the cycle. The points of maximum declination of the Moon (which is what you’re asking about) are called lunar standstills, at which point the Moon is above a point about 28.5° from the Equator.