Wave-particle duality states that a particle has both wave properties and particle properties when one is not observing it.
1) What is an observer? Need it be anything living or can other particles also act as observers?
2) When doing the electron double slit experiment--shooting just one electron at a time, the electron goes through both slits at the same time (if one is not observing it). Does that say that the electron is on every single geographical point at the same time?
The wave particle duality is a man-made simple solution to understand the properties of electron/photon. The wave-particle duality does NOT say that a particle has both a wave property and a particle property when NOT observing it.. it in fact says that to understand the properties of a particle, one must consider it to be a wave sometimes and a particle at some other times.
An observer is simply defined as something that increases the entropy of a system.
Here, you must not consider the "geographical location" of an electron. Treat the double slit experiment as a transfer of wavepacket from point A to point B.
Rox, I highly recommend that you get a copy of Richard Feynman's QED: The Strange Theory of Light and Matter. You are asking some interesting questions, but you will need to state them more precisely before you can get an answer that will be fully satisfying to you. QED is both one of the most interesting physics reads I know of on the oddness of things quantum, and simultaneously one of the most precise. Feynman wrote it for a non-mathematical friend, and avoided using any equations (well, except in some footnotes, just to brag about the truly amazing fit of complex numbers to the problem of quantum mechanics). Unlike many pieces on this subject, Feynman will not lead you astray with false or glitzy analogies. He realized that reality itself is quite, quite weird enough without any window dressing.
1) Nobody knows for sure. In the words of @JohnRennie, "most physicists are in the 'shut up and calculate' camp".
2)Yep, though with varying probability. It has a tiny probability of being on the moon, for example. But the bulk of its probability is in the vicinity of the slits. Given that an electron has gone through the slit (i.e., we have seen it on the screen--these are the only electrons we even consider in a YDSE), then the probability becomes $1/2$ for each slit, and $0$ elsewhere (barring strange effects like barrier tunneling etc).
See the "why don't they need to be close" section of this answer, for an intuitive-ish way of looking at wave-particle duality.
Note that even after collapse, the particle is still a wave--Just an extremely restricted one.
This confusion is easiest to resolve by learning the many-worlds interpretation first, and then, once everything is clear, transferring the intuition to other interpretations that require more philosophy. In the many worlds interpretation, you view the wavefunction as a giant wave. But it isn't a wave associated to one particle, or a different wave for each particle, but a wave over all possible configurations of all the particles together. The electron and the detector are both together waving through the space of all possible configurations, of all possible worlds. This is true in any interpretation--- it is just a property of quantum mechanics. The difference in interpretation is mostly in how philosophically real you say the other worlds are. The many-worlds interpretation does not discriminate between the world we see an all the other worlds quantum mechanics said we could have seen.
When you have an electron that is spread out, and it interacts with a detector, the detector is put in a different state depending on the position of the electron. The configurations of the detector are then said to be entangled with the electron, an this means that the electron is no longer as spread out, when considered relative to any one given configuration of the detector. This is the process of "collapsing the wavefunction".
The many-worlds interpretation explains the wave-function collapse completely, along the lines suggested by Heisenberg in the analysis of particle tracks in a bubble chamber. There is nothing required beyond plain old quantum mechanics. But there is an issue, in that everything stays in a superposition of all possible worlds, while our experience is of only one world. Whether this is just a property of our experience, or whether this means that quantum mechanics is not the final theory, I don't think is definitively settled. I could imagine it going either way, although I tend to think this is a property of our experience, and quantum mechanics is complete.
