How does the Sun move in the southern hemisphere?

Quite differently from the northern hemisphere...

From the North pole the Earth rotates around its axis counterclockwise, and clockwise, if we look from the South pole. So for a person looking at the sky in the southern hemisphere the sun moves counterclockwise relative to the South celestial pole.

How do sundials work in the southern hemisphere?

Markup is the same, but numbering ...

The hours numbering in the southern hemisphere will be "inverted" relatively to the noon mark. Polar gnomon should be directed to the South pole. Alas, there is no accurate marker on the celestial sphere for the South pole now, like Polar star for the Northern hemisphere. A rough guide is Octant constellation.

Can a sundial made for a certain latitude be used elsewhere?

It depends on sundial's type and the place...

The picture explains how to change the position of the horizontal sundial with polar gnomon, depending on the latitude of new location. The dial surface will no longer be horizontal, but the style will still point The Pole.