Astronomy Puzzle for Richard

I know that different times of the day, different times of the year, the sun and moon appear in different parts of the sky. I know that in the higher latitudes the sun (and moon?) never get too far off the horizon. On the equator they can be directly overhead.

But I don't know what the shape would look like, if projected over a whole year. Would it be a band? A bow tie? A semisphere?

So here's the puzzle. Is there any place on earth where the sun inhabits every inch of sky during the year?  How about the moon -- same or different spot on earth?