Yes, you can copy and paste a single layer at a time in ArcMap. If you are in Data view, select the layer you want to copy, right click, and select Copy. If you are in Layout view, you can also use the Copy command from the Edit menu. To paste, select the dataframe in the TOC you want the layer pasted into, right-click, and select Paste Layer.
You cannot copy and paste multiple layers in ArcMap.