How to scale and shift the coordinates of a Graphics object?

You can use Scale on First @ g2 (which contains the graphics directives and primitives):

Show[g1, Graphics @ Scale[First @ g2, 2, {1, 1}], AspectRatio -> 1, 
 Axes -> True, ImageSize -> 230]

Alterantively, you can use combination of Scale and Translate:

Show[g1, Graphics @ Translate[Scale[First @ g2, 2], -{1, 1}/2], 
 AspectRatio -> 1, Axes -> True, ImageSize -> 230]

or

Show[g1, Graphics@Translate[Scale[First@g2, 2, {0, 0}], -{1, 1}], 
 AspectRatio -> 1, Axes -> True, ImageSize -> 230]

same picture

I'm not saying that this is what you need for your problem, but sometimes when you want to superimpose graphics like this, you're looking for Inset:

Show[
 g1,
 Graphics@Inset[
   Show[g2, PlotRangePadding -> 0],
   {0, 0}, {Center, Center}, {2, 2}
   ],
 Axes -> True, ImageSize -> 230
 ]