Does a sunset in the west and a moonrise in the east ever happen at exactly the same time?
Yes, a sunset and moonrise can occur at about the same time. Specifically, it occurs on the night of a full moon! Let’s convince ourselves of why this is…
Whenever we think about sunrises/sunsets in relation to moonrises/moonsets, we need to think about where on Earth we are. When we can see the sunset in the west, we are on a part of the Earth that was illuminated and is now rotating such that sunlight will not shine upon us any more. When we see a moonrise in the east, it is because the Earth is rotating towards the moon enough that it has risen above the horizon. So for us to see a sunset and moonrise at almost the same time, the Sun and Moon need to be on opposites sides of the Earth. At which moon phase does this occur?
Whenever we need to think about the specifics of moon phases, we should think about how the positions of the Sun, Earth, and Moon relate to one another. For example, a new moon occurs when the Moon is between the Sun and Earth, so we are unable to see any illumination. On the other hand, when we observe a full moon at night, it’s illuminated face is due to sunlight shining on it – the Earth is between the Sun and the Moon. This scenario is exactly what we need to be able to see both the sunset and moonrise within minutes of each other!
Above is a little figure I made to pictorially represent what I described above. Now, the sunset and moonrise do not occur at exactly the same time because the Moon’s orbit is slightly inclined with respect to the Earth’s orbit around the Sun. Even so, with a little planning, you can observe three very impressive astronomical events (sunset, moonrise, and the full moon) all in one night.