Does the Sun and Moon basically appear to "move on the sky" identically regardless of location and date?

Question

For a long time, I've been trying to use Stellarium (simulation) and many visualizations and animations in order to get a good idea of how the Sun and Moon move across the sky, as seen by a human from the surface of Earth.

It's surprisingly difficult to grasp this, at least for me.

It seems to me that the Sun always "rises" in the "East", goes slowly up to "above our heads", and then sets to the "West" in the same manner, making a kind of half-circular path. After that, after some unknown time, the Moon kind of "follows" the Sun in this (more or less) exact path, also coming up from the East and "setting" in the West.

The fact that the Moon is orbiting around Earth, and Earth orbits around the Sun, and yet they move like this, makes no sense to my brain. It seems like the described path would only be true for the Sun, or the Moon, but not both.

And is it always (roughly) the same? Or does this vary depending on where you stand on Earth, and what date/year it is? It was very difficult for me to make my own experiments in Stellarium. It seemed like it would vary a bit, but not too much.

The reason I'm trying to figure this out is both because of general curiosity, and wanting to accurately describe the Sun and Moon in my story. In The Lord of the Rings, Tolkien frequently talks about the movement and look of these objects, which appear to follow the same rules as in our world. It's clear that he had very deep knowledge of how these things appear to move on the sky over the day, and I like such "details" to be accurate.

Basically all the animations and illustrations I find of the movement of the Moon and Earth around the Sun are focused on a "space overview" rather than how it looks from Earth.

Answer 1

The Moon's one month orbit makes the Moon rise about 50 minutes later each day for people who aren't at extreme latitudes. (I'll examine extreme latitudes later). The Moon's orbit is inclined by 5° with respect to the ecliptic. This means that the new Moon more or less follows the path of the Sun that very day, while the full Moon more or less follows the path the Sun took six months earlier. This is why the winter full Moon rises so high in the sky in temperate latitudes.

What about more extreme latitudes? The Amundsen-Scott South Pole Station is the epitome of extreme latitudes. At the South Pole, the Sun sets six months after it rises while the Moon sets two weeks after it rises. In winter, the moon rises about a week before full Moon and sets about a week after full Moon. In summer, the moon rises about a week before new Moon and sets about a week after new Moon.

Answer 2

The motion of the Sun and Moon from the point of view of one on the surface of the Earth is almost entirely due to the rotation of the Earth

The Earth rotates once a day (*) so the sun, moon, planets, stars all seem to rise in the East and set in the West.

THe Sun, moon and planets all appear in the same line, because to an approximation all are in the same plane. To a first approximation all the planets, moon and sun are in the same plane and so appear in the same line for one on Earth.

Over time, the movement of the Earth, moon and planets cause the sun, moon and planets to move relative to the stars. The moon moves relative to the Earth. It moves relative to the stars, and relative to the sun.

The planets move around the sun, the movement (relative the Earth) is complex.The planets backward each time that the Earth overtakes the planet (as earth is overtaking mars as I write.

These position in the sky varies due the differenes in latitude