To get more precision:
- You can define a snap point in menu
Insert > Insert Snap Point/Line and enable snap to snap lines in menu View > Snap Lines > Snap to Snap Lines. When you drag the Pivot Point with the mouse, it will snap to the snap point.
- You can use the
Position and Size dialog (key F4) and enter the exact position of the Pivot Point and in addition the value of the rotation angle.
The Pivot Point is not restricted in its position, you can drag/set it where you want. But the setting is not sticky and you have to do it before each rotation.
You can get a series of copies with the tool in Duplicate in menu Edit > Duplicate. The tool has means to alter position, size and color for the series.
You should get an identical copy of a shape with a simple copy+paste, but that was erroneously changed to a little shift in version LO 5.3, it is back to past at original place in LO 5.4. In LO 5.3 you can use the duplicate tool for that.