I don't understand, why can't a fractal be largely evenly distributed at the "zoomed-out" level?

Because it follows, from the definition of a fractal, that it 'looks similar / the same' at every 'level of zoom'. It should still look like a fractal at the zoomed-out level.

Am I allowed to use my intuition here? Position and speed are entirely different concepts or dimensions. By the time I've determined a position, my action has let the particle go ... too late to get its speed. If I measure its speed, by the time I've tried to pinpoint its position, its gone. I wonder if that's a correct observation as I'm using a lot of undefined words here?

It's not that intuitive. Lets look at the problem in one dimension. x(t) and vx(t) being the position and the speed along this one dimension. Lets say you have an object, and you measure it's position, then you have x(0). If you measure it again , you'll have another position (x(1)), from which you can approach dx / dt = vx. Measure it again (x(3)) and you can approach the acceleration (d²x/dt². If we consider that in classical physics, we typically don't use d^3 x/dt^3, we can calculate all three of these exact.)

Note that we need TWO measurements (at diffrent times) (to find x and vx, ignoring ax here).

In quantum, you can find the position and momentum (=velocity) in one simultaneous measurement. However, this measurement will ALWAYS have incertainty inherent to it. There's no possible way to bypass this. This is because you measure the wave functions that are applicable to this position. These tell you the impulse (velocity), but not exact, because they are wavepackets.

Is this acceptable:

Everything has both because an existing particle must BE somewhere, at least for a moment. A wave means that an existent particle is not an island, but has influence on its surroundings. The WAY it influences is it "waves at you."

Whoops. You just uttered a couple of curse words. What is "mass"? Is it the "resting" content of what the wave-particle is? What is "charge"? Some special kind of wave-particle influence? (Since there appear to be difference kinds of charges, these wave-particles must be different. If that is so, we must get inside them to explain the differences.)

I'm talking about mass and charge as proporties of the particle / wave (... insert more advanced theories as to why that is not 100% accurate.) Mass for gravitational interaction, charge for electromagnetic interaction.

Quantum says a particle is in multiple places at once (everywhere it's wavefunction says where and howmuch probability to find it) until you 'probe' it. Probing it requires interaction (photon / electron / electromagnetic field / etc.) Without interaction and a probe, we cannot measure. By probing, measuring, we force the object into it's particle sense, much like the video said. The wave is everywhere, but the particle is only in one place. But before you probe it, you have no way of knowing which place this will be, and some experiments show it's actually waving through the place (interference / diffraction) . This is why you only require one measurement. One measurement tells you where the particle is, what wave functions still apply, and by the wave functions you know it's energy and thus it's velocity. (Altho, since you go by waves, you'll never know it exact.)

Question: to separate the particle from wave, could we say the particle part is what we see for position (corresponding to mass), while the wave part is the moving part (corresponding to energy)? Maybe that's too strong a statement.

Wrong, and you can't truly seperate particle from wave. Both particle and wave have their own properties, but they're connected. Both particle and wave have mass, impulse, position. Particles are entities, they cannot be split. A photon is absorbed as a whole, or not at all. They are 'quanta' of energy. Yet these unsplitable quanta are in multiple places at once - as waves. Even when you probe a particle, and it thus becomes a particle, it still has it's wave functions. These will just be much more located, much less probability. A single wave packet, rather than a superposition of lots of them.

Kind of like, "It's a probability because I don't really know and can't know, so I label it by saying: probability = I don't know. Here is a concept I can pose: Suppose the observer (God) as mind only were outside the physical universe. Then since a particle exists with existence properties, it would HAVE a position and speed at the same time. It's just that only this "God" would know it.

I don't know, I truly don't. I'm inclined to say no, the particle has multiple positions and velocities each with their own probability. It only gets one position and one velocity once you probe it. Schrödingers cat, the cat is both dead and alive, both with their own probability. You do not know which until you look. In quantum mechanics, (I know this isn't very intuitive), the cat is both alive and dead until you look. Once you look, you force the probability waves (% alive, % dead) into one particle with one position and momentum (100% alive or 100% dead). Reality only becomes what it is once you look. Once you probe, before that, all options are there, and likely along their probability.

I believe experiments have demonstrated this probability. I'm wondering though if this is for a large stream of electrons rather than one. I suppose so.

The 1st link is okay, but not the 2nd. Why would the electron hitting the wall with particle-ness cause the wave part to disappear everywhere else? Is that because it's one electron alone whereas sound and water waves involve a medium of sound and water particles? This is makes water and sound misleadng. If ONE electron travels, does its wave appear in 3-dimensions everywhere or is that only for a stream of electrons with a probability distribution? Or does the single electron carry its waveness with it only in the direction which it travels?

It's a mixture of thought experiments and real experiments but they're pretty sound. To see the duality, you should look at the double-slit interference.

http://en.wikipedia.org/wiki/Double-slit_experiment
Wall of text incoming, but i'll try to make this as clear as possible. It's not that easy to understand and it feels very illogical.

The two slit experiment shows interference between photons. (Same has been done for electrons!). As waves, this experiment is easy to understand. Lets look at it in terms of particles. Imagine we have a stream of particles, and each particle has a probability to where it'll end up, then you could find the same results...

However : Imagine you attach a measurement to see which photon goes through which slit, and then -while the experiment was double-slit- you only plot the ones that went through slit1. You'd get a normal single slit pattern. However, we can do this for both slits, and the total of the experiment is just the adding up of our slit 1 and 2. However, this won't give us the clear interference pattern, yet something entirely diffrent! Measuring which slit our photons go through changes our experiment and ALTERS REALITY. In the case of not measuring (where photons remain waves of probability), it's impossible to know whether they go through slit 1 or slit2. There is even to pose more, the particle goes through both slits at once. That's exactly what the second video is showing, your waves can pass through both holes at once - as waves!- and will only become a particle with a single position and momentum once it is probed (in this case hits the wall)

To go back to your question. ONE electron travels everywhere by huygens principle as wave. Until you probe it, then it appears in one place, with one velocity. Before you probe it, it is everywhere, and you have no way of knowing where it'll end up. That being said, the wave functions follow certain laws. (Schrodinger equation). Imagine an electron is being pulled into a positive charge (electromagnetic force), then obviously the probabilities of the wavespackets aligning with the charge moving there are much higher than those where the wavepackets move away from the attractive force. However, since there is uncertainty concerning the particles velocity inherent to quantum mechanics, what if the particle has enough velocity to escape the attraction? Despite it being more likely to find the particle close to the attractive force, it is still possible to find it moving the other direction, because it had the energy to escape the potential well.