Before thinking about circuits, let's think about two conducting spheres of charge that I connect by a wire. Before I connect the wire, sphere 1 is at voltage $V_1$ and sphere 2 at voltage $V_2$, let's say $V_1>V_2$. I find it useful, in terms of thinking about what's going on, to notice that if the spheres are the same size then saying the spheres have different potentials is equivalent to saying that the 2 spheres have different charges residing on their surfaces (you can justify this by noting that the capacitance of a sphere is determined by its radius).
Now let's connect the spheres. What will happen? Well, a current will flow in the wire. This will take positive charge off of sphere 1 and deposit it on sphere 2 [strictly speaking if you want electrons to be charge carriers, then negative charge is flowing from 2 to 1; but in terms of thinking about what's going on it's easier to imagine, and mathematically equivalent to say, that positive charges are going from 1 to 2]. This in turn changes the voltages on the two spheres; $V_1$ decreases and $V_2$ increases. The process stops when $V_1=V_2$. Again, if the spheres are the same size this condition is equivalent to the charges on both spheres being equal.
OK, now imagine a battery hooked to a resistor and a switch, the simplest circuit imaginable. Before we close the switch, terminal 1 is at $V_1$ and terminal 2 is at $V_2$. At this point, it makes perfect sense to think of each terminal of the battery as being a sphere of charge. Then we close the switch, this is like connecting our spheres with a wire. Based on our silly model of a battery, you would the voltage between the two terminals of the battery (ie, $V_1-V_2$) to decrease until eventually it reached equilibrium with $V_1=V_2$. Clearly, a battery does not behave like two spheres of charge after the circuit is closed.
The whole point of a battery is that it maintains the potential difference between its two terminals. After we close the switch, a little bit of positive charge flows from terminal 1 to terminal 2 by going through the circuit. Naively this means that terminal 1 has less positive charge and terminal 2 has more positive charge, so terminal 1's voltage decreases while terminal 2's voltage increases. Inside the battery, some process takes place to to take the excess positive charge on terminal 2 and put it back onto terminal 1. Whatever this process is, it cannot be electrostatic, because positive charges following the electric field can only ever move from terminal 1 to terminal 2 [positive charges move from high voltage to low voltage, if the only force is electrostatic].
The details of what the battery does to maintain the potential difference varies depending of the kind of battery. A conceptually simply example of a battery is a Van de Graaff generator. In a Van de Graaff generator, you have a conveyer belt that literally carries the excess positive charge on terminal 2 and deposits it back on terminal 1, undoing the naive 'equilization process.'
Most useful batteries rely on some chemical process to maintain the potential difference. For example, one can use oxidation reactions to do this. The details involve some chemistry (there's a wikipedia summary at http://en.wikipedia.org/wiki/Electrochemical_cell), but essentially you put each terminal in a bath of ions, and the chemical energy of the reactions at each terminal [balancing oxidation and reduction] forces ionized atoms to carry electrons from terminal 2 to terminal 1.
Conventional current and flow of charge carriers in a circuit are two different things.
A charge carrier is defined as a particle carrying an electrical charge through a conductor or semiconductor. What these charge carriers are can differ as for metals, these charge carriers are usually electrons. Therefore, current in the circuit is determined by the flow of electrons(i.e flows from negative plate to positive plate as you suggested). On the other hand, in semiconductors, the charge carriers can be positive holes(and in that case the current direction would be from positive plate to negative plate). Charge carriers of a circuit can also be both positive and negative at the same time.
Due to the fact that flow of charge carriers can be in any direction, the direction of current is usually specified in the circuit with an arrow. Thus, $I$ has a positive value if its direction is the same as the arrow(see below) and a negative value if its direction is opposite.
However, the term conventional current can also be used. The direction of conventional current is assumed to be from positive terminal to negative terminal although the flow of the charge carriers may not be in the same direction. This is not so important for technicians since the direction of $I$ doesn't affect its value. If we are not working with circuits, however, that is a different story.
Best Answer
The experiment calls for noting when the foil just lifts off the bottom plate. Since the foil and the washer are both presumably good conductors, the charge density should be the same until they stop touching. So the "goodness" of this assumption depends on the variation in the voltage across the capacitor after the foil lifts off.
The foil should feel forces from both plates, so I think your source is wrong in this. Horizontal forces come from the fact the plates are not infinite, and therefore some of the field goes beyond the plates.
Washers are both readily available and fairly good conductors, so they're used in a lot of "do it at home" type experiments.