How many cookies in the cookie box? -- Tiling stars
Let me answer your question partially for the hexagram case.

You can make the following tiling

$\hskip1.2in$

By this you will cover 12/14=6/7 of the plane (count the triangles in the dashed quadrilateral).

Is this optimal? I would think so. Although I am not giving a proof I will provide some arguments. One can ask, how good we can fill the space (triangle) in between the pointy spikes. In the above tiling we fill half of it. Can we do better?

$\hskip20mm$

It is possible that two hexagram will intersect this space, but then they cover very little of its area (without proof). If there is only one hexagram intersecting I assume that its tip touches the concave corner of the other hexagram as depicted in the picture. If this would not be the case we can improve by moving the intersecting hexagram to this corner (again no proof here). Under these assumptions it is not hard to see that the case where there is side-to-side contact maximizes the intersection. If you do the math, then you will find out that the area of the intersection equals 
$$\frac{\sec ^2(x)}{2 \sqrt{3} \tan (x)+2}.$$

The plot of this function looks like this and shows that our intuition was right.

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