So far as rising in the East and setting in West, yes. Otherwise, kinda. The Moon's orbit is inclined to the ecliptic from about 5.0° to 5.3°. So it will rise and set nearly where the Sun does so,...
There really isn't any relationship between sunset and moonrise. The Earth turns in 24 hours, so the Sun rises about every 24 hours. But in between, the Moon is orbiting the Sun on its own, so the...
The Sun does what it does best; shines fairly constantly, and proceeds in its lazy 220-million year orbit around the center of the galaxy. The Earth orbits the Sun, and the Moon orbits the Earth. If...