Astronomy: Is there a site which shows the Moon's position for a given time and latitude throughout the year?

There are a number of celestial object viewing sites that can point out the moons location.

I use a program called 'Clestia' , a 3-D interactive of the cosmos.

You can even tool around the moon, get up close to a number of craters and named features. Zoom out and see where it is relative to earth and the solar system.

That south north variation is the fact that the moons orbit is tilted 5.1 degrees from the ecliptic.

Yes

The US Naval observatory have a site

The Moon's Orbit has a 5 degree Tilt from the Ecplitic

The moon's orbit is tilted by 5 degrees compared to our equator, so it appears to swing north and south.

https://www.psi.edu/epo/resources/inmotion/lunar_o...

Yes, try this:

https://www.moonpage.com/

Any time, any year...

Yes. The U.S. Naval Observatory has a web page that will calculate the altitude and azimuth of the Moon or Sun every 1to 10 minutes for a particular location and day of the year. The location does NOT have to be in the United States.

The altitude of the Moon is the reverse of the altitude of the Sun be cause the Moon's orbit in tilted at 5.12° degree angle to the apparent orbit of the Sun. . the Moon's altitude on the meridian at full Moon is highest in December and lowest in June, the opposite of the Sun at LOCAL noon.

Other countries often have a similar web page.

https://aa.usno.navy.mil/data/docs/AltAz.php.

The earth is tilted in it's orbit which causes the seasonal change. If you have noticed, the sun changes position during the year too as the earth moves around the sun. However, the moon is in the same plain as the sun and so changes it's apparent orbit the same way as the sun. This is most noticeable during the equinoxes in March and September when the changes move faster.