Does the Sun and Moon basically appear to "move on the sky" identically regardless of location and date?
Like Fred said, this probably belongs on Astronomy SE, but I'll give my best answer anyways.
No, the sun and moon do not take the same path each day. The key to understanding this is knowing that the earth is tilted 23.5$^\circ$.

Let's take an example- the arctic circle. On the summer solstice, the sun doesn't set- It goes around and around (called the midnight sun). In the winter solstice, the sun doesn't rise in the Arctic circle.
I think a careful observation of mine might be useful for you. The word Mediterranean means &quot;middle earth.&quot; Tolkien is likely referring to the specific geography- specifically the area around the Mediterranean Sea. In fact, Tolkien does state that the Mediterranean is the place he envisioned, but he does use a flat earth, so the cosmology doesn't really match up with reality. This is also another nail in the coffin for flat earthers.