It's definitely a real thing!
Think of the double slit experiment, the typical example where pretty much the whole the of quantum mechanics becomes manifest. I am sure you've heard of this experiment before, so I am not going to describe the apparatus.
Basically you have a gun that shoots electrons (even one at a time) at a screen, that has 2 slits.
There are of course three possibilities for the electron trajectory:
1) The electron hits the screen and does not get through the slits: it's either absorbed or reflected - either way, we don't care, it's gone;
2) The electron goes through slit 1, after which it will have a wavefunction $|\psi_1 \rangle$;
3) The electron goes through slit 2, after which its wavefunction will be $|\psi_2 \rangle$.
Let's focus on what's going on beyond the screen, taking into account only the electrons that have passed through the slits.
Since the electron must have gone through one of the two slits, but we have no way of knowing which one without performing a measurement, the total wavefunction can be written $|\Psi\rangle = \frac{1}{\sqrt{2}}( |\psi_1 \rangle + |\psi_2 \rangle) $,
where the $\frac{1}{\sqrt{2}}$ factor is just for normalisation. So the electron is in a superposition of $|\psi_1 \rangle \;\&\; |\psi_2 \rangle$.
Now, even outside the context of Quantum mechanics this expression makes sense: if you don't know which slit the electron went through, you'll just assign a 50% probability to each possibility and keep them both.
But Quantum Mechanics kicks in.
Both states will have a time evolution, which causes the wavefunction at the slit position (basically a delta-function since you know that the electron must be there) to spread.
If the electron is at either one or the other slit, then only one delta-function spreads and becomes a Gaussian-like wavefunction after a while, getting larger (more spread out) and larger.
If the electron is at both slits though, you'll have both delta-functions spreading to Gaussians. The point is that the Gaussians, when they start to superpose, will interfere! The Gaussians here are wavefunctions, which obey the Schrodinger equation (the wave equation- hence this approach to QM is called wave mechanics) and therefore display inteference!
So you end up with the well-known interference pattern. But it's essential that the electron is at both slits, i.e. that its wavefunction $|\Psi\rangle$ is in a supersposition of $|\psi_1 \rangle$ and $|\psi_2 \rangle$.
Here's a sketch:
Mathematically, it all boils down to the fact that the inner product of the superposed wavefunction gives you extra (interference) terms: I'll go into the details if you want but I think that you should really get the physical picture first.
To answer your questions:
1) Superposition is a property of every solution to the Schrodinger equation, which is a linear partial differential equation: every two (or more) solutions to the Schrodinger equation can be combined together to form yet another good solution. In this sense you might say that it is "obeyed".
2) As shown in the example above, superposition gives rise to observable and measurable phenomena, such as the interference pattern of particles.
3) I wouldn't know what you mean by behaviour, but if you mean it as something that is exhibited by the the wavefunction, sure, yes.
