To copy, press Ctrl+C. To cut, press Ctrl+X instead. Then move the mouse to the top left corner of the area to copy/cut. Press and hold the left mouse button while dragging the mouse to the bottom right corner of the area to copy/cut.
To paste, press Ctrl+V. Move the mouse to position the clipboard contents (it will snap to multiples of 4 pixels), then click to place it. Before placing it, use Ctrl+R to rotate it, Ctrl+shift+R to flip it horizontally, or Ctrl+arrows to move it by 1 pixel at a time.