Can the moon appear to be in any position in the sky?
It seems to me that the Moon has appeared in many locations - apparent locations - in the 'dome of the sky'. I know that the Sun follows an arc from east to west and that the arc is smaller in winter than in summer. Therefore the Sun cannot appear in every location in the 'dome of the sky'.
My question is: can the moon appear in any (every?) location in the 'dome of the sky'?
Yes, but only if you don't constrain the time or position on Earth you have to be at in to see the moon at any particular position, and if you only measure the position relative to the surface you are standing on, as opposed to other objects in the sky.
I believe we can assume that the question is meant in the context of being at a fixed position (by mentioning winter vs. summer - there is summer at some place at any time in the year, after all ;) ).
Like the sun, the moon, from our perspective on the surface of the Earth, rises in the East and sets in the West. However, it does not rise exactly due East and set exactly due West. If you were to track the position of the moonrise along the horizon over the course of a year, you would notice that is varies considerably. It cannot, however, be located everywhere in the night sky, even over the course of a year. To understand why, we need to understand some celestial geometry.
The ecliptic plane is the plane which a hypothetical line between the Earth and the sun sweeps out over the course of a year. In the image above, you can see
how the Earth's orbit exists within this plane.
Also notice that the line from the Earth's geographic south pole through its north pole (its rotational axis) is not perpendicular to this ecliptic plane. In fact it is 23.5 degrees off this perpendicular direction.
Notice that the Earth's rotational axis points in the same direction regardless of which side of the Sun the Earth is currently on. This rotational axis always points towards the star Polaris. This means that on different parts of the year, the sun will rise at varying points along the horizon.
The Earth's equatorial plane is a plane defined to extend out from the Earth's equator in all directions. This plane is inclined relative to the ecliptic plane by 23.5 degrees (the same amount as the rotational axis is tilted relative to the perpendicular direction of the ecliptic plane).
Now consider the Earth moon system. In order to determine how far North or South the moon can appear to rise from, you need to consider the angle between the Earth's equatorial plane and the moon's orbital plane. Since the Earth's rotational axis is inclined 23.5 degrees relative to the ecliptic, and the Moon's orbit is inclined 5.14 degrees relative to the ecliptic, the highest the Moon's orbit can be relative to the equatorial plane of the Earth is 28.64 degrees.
Thus, the Moon cannot be directly overhead of any point on the surface of the Earth if that point is 28.64 degrees above or (by an symmetric argument) below the equator.
Thus, the Moon cannot appear in every location in the sky for a given location on Earth. There are parts of the sky that the geometry of the Earth moon system simply will not permit the Moon to exist.