Newbie question: Atom identity. How can you talk about two electrons if electrons are identical? How can you talk about two electrons if they are identical (indistinguible)? Does it make sense to let an electron to have an identity by itself?
If they are on diferent places the place they are is a diference (they are not identical). 
The act of labeling, naming them make them distinct. If I say let A be an electron and B another one, then they are distingible by name...(they are not identical)
Edit: if electrons are indistinguible by principle, then you cannot name them properly. A tick after you name one of them it could be any other electron in the universe or not exist anymore. You could say let this be electron A, but a tick after, A means nothing. The most you can say is at some point in time there were A and that "electronism" would remain, but this "electronism" would emerge in diferent form as long as "electronism" holds.
I think that the indistinguibility property makes the interpretation of a particle who has an identity and live forever hard to follow. One need to create strange ideas like "it follows all the paths" or "there are several paralel universes"...
 A: We can do an experiment that distinguishes a box with two electrons in it from a box with just one electron in it. Just measure the electric field near the box and the field near the two electron box will be larger.
We cannot do an experiment that distinguishes between a box with electron A on the left and electron B on the right from a box with the electron A and B swapped. This is what indistinguishability means.
A: This is actually a very deep question, good job!
If you think about what counting means, it is a sort of song and dance game, where the song is usually a sort of chant that we have just agreed upon, where the important thing is that each sound that you chant is different from every other sound in the sequence. Then the game has certain rules, where in time with the chant you are pointing at objects, and you must never point at the same object twice, and you must point at all of the objects, and then whatever the last word you chanted was, we say is the count of the thing. 
You are right to intuit that this game cannot be played in this form with electrons: at least, not according to quantum mechanics. You can point at distinct states that the electrons are occupying, and you can count those, but you cannot count the actual electrons, because they are indistinguishable particles, and there is generally a non-zero probability that they switched places, and this probability actually has some measurable consequences, in that the electrons behave statistically according to Fermi-Dirac statistics.
With that said, the fact that there is no observable difference between any electrons, actually kind of helps us out a little bit. It means that we can speak very loosely about a state occupied by an electron, as “an electron,” and not worry about tracking the electron in that state to see if it changes places with another electron. The only reason we'd really care about them changing places, is if it made some sort of difference to some sort of measurement, and it cannot.
So when we count electrons, or try to label them, we are really labeling the state that they are in, and counting the states that they are in, and confirming that those states are occupied by an electron. But at a deep level indistinguishability does mean that we can't tell you that “it's for sure the same electron occupying this state at time $T+1$ as had been occupying it at time $T$.
Of course, the states occupied by electrons, do have real physical consequences. For example, you can measure the electric charge in a box, and count electrons that way. Or for another quick example, water contains these atoms of hydrogen and oxygen, which if the electrons were separated and not configured so that the electrons were in these shared states between the hydrogen and oxygen atoms, would be flammable. But they're not once they are water.
A: You can label billiards balls, ping pong balls, marbles and even BBs. But there is no way you can label electrons, even in principle. Think about 2 identical electrons in a head on collision. Before the collision they move in opposite directions along the same straight line. After the collision, 2 electrons move away from each other at some angle to the initial direction less than 180 degrees. How would you be able to tell which electron was which in the final state?
The answer is you can't. So you have to consider the final state as a superposition of both possibilities. Exchange degeneracy.
