The ground is connected to every point of the metal plate by the metal wire and the metal plate.
I mean, there is a metal wire that connects the ground to the plate, and then there is the metal plate that connects every point on the metal plate to each other, therefore every part of the metal plate is equally connected to the ground.
When you touch the plate with the grounded wire, electrons move to some direction through the wire. As every point on the metal plate has exactly the same voltage, it does not make any difference what part of the plate you touch with the wire.
Metals consist of small crystals; within each crystal exists the "free electrons" which are shared by all of the atoms in the crystal lattice. The number of free electrons per atom depends upon the details of the atoms, but is most often 1 or 2. The free electrons are visualized as "the electron sea" in the Drude model, devised ~1900, and is semi-classical. Introductory condensed matter texts often start with this model. The situation is slightly more complicated with alloys, but the same ideas hold.
In the electron sea the electrons are electrically shielded from each other by (a) the net positive charges of the ion cores and (b) the uncorrelated motions, essentially random, of the free electrons, which are described using the kinetic theory of gasses.
The crystal boundaries serve to impede the free flow of these electrons from one small crystal to the next, and also serve as scattering sites which continually randomize the motions. The velocities of the free electrons are quite large. When an external electric field is applied, it appears as a net "drift velocity" in the electron sea. This is the current in that piece of metal.
When you bring two clean pieces of metal together all of the above is still true, but there is an additional restriction: each crystal has an effective "crystal voltage" on its interior, and for crystals of the same type it should be the same. But when different metals are joined, the difference in the crystal voltage causes a voltage drop when going in one direction, and an increase in the other. This voltage difference is known as the Seebeck effect, discovered in 1821. Since the internal voltages change slightly with temperature, it is possible to measure temperature change electrically; this is the physical basis for the thermocouple.
So adding additional metal increases the total resistance of the circuit, depending upon the resistivity of the additional metal, its dimensions, and other properties.
The current is the net flow of electrons; each individual electron barely moves, but the effects are passed down the line. With alternating currents for every move forward, there is a corresponding move backward -- hence no net motion from the electron drift at all.
Best Answer
If you have a complete circuit, every piece of metal will gain and lose the same number of electrons and will not have a net charge. If you connect two plates, one to each end of a battery, the battery will take charges from the plate connected to the positive terminal and send charges to the plate connected to the negative terminal until the voltage between the plates equals the voltage of the battery. At this point, no more current will flow, but each plate will have been given a net charge.