For your first questions:
Observables correspond to hermitian (or self-adjoint) operators.
As such, the eigenvalues are real, and these values are the possible outcomes.
Also because of the hermitian/self-adjointness of the operator, when you have eigenvectors with different eigenvalues, the eigenvectors are orthogonal.
So you know the values, but what happens? The shortest story is that a measurement projects the vector onto an eigenspace, and you get the eigenvalue corresponding to that eigenspace, and the relative frequency of getting different results is the relative squared-length of the projections of the original vector onto the orthogonal eigenspaces. And any book will tell you this. To visualize this, you can imagine projecting onto an eigenspace, and then rescaling to be unit length again (so this is not a rotation). And imagine that when you do this you get the eigenvalue (somehow) as a result, and then you can imagine that it happens probabilistically in some way with a frequency equal to the ratio of the post-projection squared length over the pre-projection squared length.
But you wanted more. How does measurement relate to unitary evolution? Two parts remain the same when you look at the process realistically. First, the ends results are indeed orthogonal. Second, the experimentally observed frequency is equal to what the post-projection squared length over the pre-projection squared length would have been had there been a projection. So now let's look at what happens.
What happens is that the system evolves according to the Schrödinger equation. And we call something a measurement when it evolves into a sum of orthogonal states, that will always and forever more remain orthogonal. A common way to achieve orthogonality is to have no spatial overlap, for instance a stern-gerlach device can deflect the single beam into two nonoverlapping beams. Wavefunctions are in configuration space, and configuration space is staggeringly vast, so once those beams start to interact with large numbers of different particle to make them move differently, the wavepackets are incredibly unlikely to ever overlap again. This is a prerequisite to calling an evolution a measurement.
The other thing you need, is to have those wavepackets be (at some moment after they become forever orthogonal) eigenvectors to the observable. So for instance, in the stern-gerlach, the spin (bi)vector for the two spatially nonoverlapping beams needs to become polarized, all spin up in one beam and all spin down in the other beam. How does this happen? Well, the Hamiltonian for an inhomogeneous magnetic field does this naturally, an example is available at [http://dx.doi.org/10.1119/1.4848217](this nice article in the American Journal of Physics), [http://arxiv.org/abs/1305.1280](arxiv version). If you don't want to read the article, the punchline is that the single beam bifurcates, and a volume goes one way and a volume goes the other way, and the relative volume depends on how much the original state had of the up and down, and all the volume that goes one way has the spin polarize one way and every part of the volume that goes the other way has the spin polarize the other way. This is literally how you get the repeatability of identical measurements, and the relative fractions. All from the Schrödinger equation.
This is how a measurement happens, and this allows weak measurements too, which are often what actually happens, and sometimes is what is desired. Plus it's what is described by the actual evolution equations. And is what is observed by the lab, and it requires that you actually interact with the system to do a measurement rather than wave your hands and hope a measurement happens.
But what about the probabilities? When the beams deflected and polarized, the squared norm of each deflected wave is equal to what the post-projection squared length over the pre-projection squared length would have been had there been a projection. So that part is, again, already taken care of by the Schrödinger equation.
But since all the beams exist according to the Schrödinger equation it might look like a measurement hasn't happened. After all, some of the beam went left, and some went right. But not only are the beams orthogonal, they must always and forever more remain orthogonal, which actually requires that each now acts like the other one doesn't exist. Each wave is in configuration space, the configuration space of every particle in the entire universe, so the universe now has two parts, each of which acts as if it were all by itself, it is exactly that sense in which the measurement has "happened" with different outcomes. There are now parts of the wavefunction that all act independently. And there is no harm done if the people in each part (whose particles are part of the configuration space) each decide to act as if the other parts don't matter. So at any point they can disbelieve the existence of the other parts, and it will not contradict anything about the evolution of their part of the wave.
This is exactly why I described ratios of post-projection and pre-projection. If your want to rescale your part because it will never be influenced by the other parts, it changes nothing. The overall norm affects nothing, only relative magnitudes matter, and even then it only affects the math if the waves are not orthogonal. So at some point you can rescale (or not), and at some point you can act like a measurement happened (as long as they are now orthogonal, will remain orthogonal forever more, and were at one time in separate eigenspaces).