To be clear, you changed the “drawing axes” with the AxesTool and saw no effect externally in Google Earth ?
If the above is true, you need to be aware that the “drawing axes” is a UCS (user coordinate system) that does not actually change the model origin and axes. (Right-clicking the drawing axes brings up a context menu that has a Reset command that will set the axes back to match the model’s axes.)
You can rotate the model in SketchUp by changing the NorthAngle. By default the Y-axis points North. But this can be changed. Trimble has a Pro plugin called "Solar North". (Get it at the Extension Warehouse.)
If you do not have Pro, then Jim Foltz posted a cute little utility here …
Google Earth (and the KMZ / DAE export) I believe uses an Attribute Dictionary named "GeoReference".
Sometimes it’s necessary for you to tweak it. So you’ll need Aerilius’ Attribute Inspector plugin.
(Get it at the Extension Warehouse.)
You can move the model’s center point to the model’s Z_AXIS (directly above the ORIGIN,)
via the Ruby Console, by entering these 3 4 statements (on 3 lines) …
model = Sketchup.active_model
pt = model.bounds.center; pt.z= 0.0
model.entities.transform_entities(pt.vector_to(ORIGIN),model.entities.to_a)