How does one import a layer (with all its properties) from one AutoCAD drawing into another?

On the menu bar, go to Format, then select Layers.

In the Layer Properties Manager, double-click the layer that you want to copy. This will make it the current layer. Highlight all other layers and click on the little sun icon. This will freeze all of the other layers and inhibit them from being copied.

Go back to your drawing, Select All (Ctrl-a) and then Copy (Ctrl-c).

Open up your new Autocad drawing and Paste (Ctrl-v) into the document.

It can also be helpful to copy with base point if you are pasting into an existing drawing. Instead of Copy above, do Copy with Base point (Ctrl-Shift-C), then click your mouse on an a reference point (corner, centroid, etc). After you Paste (Ctrl-v) into the other drawing, it will not paste until you click someone in the new drawing, preferably on the same reference point from which you copied and the layers should overlay each other.