What is the purpose of CB radio ground plane?
Will metal mesh work for this?

Yes if the holes in the mesh are significantly smaller than the wavelength of the signal (for CB at 27 MHz that's 11 meter) the mesh will appear like a solid metal plate to the signal.

Does this mean the ground plane is actually also electrically grounded?

Yes and the ground is your vehicle's chassis.

In a stationary situation you would indeed use earth ground. You could use a conductive pipe stuck in the earth.

my antenna base has a rubber pad on the bottom

Indeed it has no electrical connection there. The ground/earth connection will come from the receiver or via the shield of the cable to the antenna.

I think you have right ideas about this.