The zero ground potential is just a reference point of common agreement, it is zero in the context of electrical circuits, in comparison with known and controlled voltages of electrical components. But the ground itself is not zero: if you put the 2 voltmeter probes in the ground, you can observe rare random voltage fluctuations.

The Earth can be thought of as an anode, and the Sun - as a cathode. But the Earth as an electron source is obviously different than a van de Graaff generator, hence possible differences in current it produces.

You won't be able "to run around charged for weeks", because as soon as you contact a conductive surface or +charge careers, you drain the accumulated surface electrons.