Is there any other movement of the fluids of a plant?

Yes, there is a true circulation in the interior of all the cells.

Does a tree form a fresh ring of wood every year?

This depends upon the character of the tree. Some trees take many years in forming it.

How is it that the layers of our Gum tree are not equal in thickness?

That is influenced by the season. One year it was very cold or dry, and the tree made little wood. Another season, the rain was abundant, and the sun did his work well, and so the concentric or ring-like layer was fuller.

But I see that on one side of the tree, the bands are thinner than on the other.

That was the side that received the cold wind. The greater the number of leaf-buds above a part of a stem, the greater the diameter.

How does the bark manage when the tree forms new wood, and presses against the liber?

The outer bark cracks and peels off. When the bark grows, it does so internally; a new liber appearing adjoining the layer of new alburnum.