How can the reduction postulate be removed with the other postulates of QM still leading to correct predictions? In the axiomatic presentation of QM, I've seen it stated many times that the reduction postulate is not needed and/or incorrect, and could be gotten rid of.
However, without the reduction postulate, wouldn't tests of QM simply yield wrong results? If I measure a system that is in some linear superposition $\psi=\sum c_i |e_i\rangle$ of eigenstates of my observable $O$, the measurement postulate says I will get one of the eigenvalues as a result, with probability $|c_i|^2$. But now, since I no longer have the reduction postulate, I will calculate that as a result of the interaction with my measurement apparatus $\psi$ evolved linearly and unitarily into $\psi'$, and therefore still consists of (possibly a different) linear superposition $\sum c_i' |e_i\rangle$ of eigenstates of $O$. Therefore, again applying the measurement postulate, I calculate that I will now get $|e_i\rangle$'s eigenvalue with probability $|c_i'|^2$. After a sequence of $n$ such measurements, close together in time, the probability that I will get some particular $|e_j\rangle$'s eigenvalue in all the $n$ measurements is $(|c_j|^2 * |c_j'|^2 * ... * |c_j^{(n)}|^2) << 1$.
However, we know from experience that making $n$ close in time measurements of the same observable will yield the same value with a probability of $1$ (or very close to $1$).
So I don't see how the reduction postulate can just be dropped without either QM giving incorrect predictions, or without modifying some other postulates somehow?
 A: You may wish to look at http://arxiv.org/abs/1107.2138 (published in Phys. Rep.; extremely long:-(, but you can look at their introduction and conclusions; or you can look at their previous much shorter article http://arxiv.org/abs/quant-ph/0702135)
They consider a specific model of measurement and show how the Born rule and the projection postulate can be derived as approximations in some cases from unitary evolution. You are right that unitary evolution, strictly speaking, does not allow definite outcomes of measurement. The reason why this is not in contradiction with experimental results is shown to be the same as the reason why reversibility of Newton's laws (or unitary evolution of quantum mechanics) is not in contradiction with practical irreversibility of thermodynamics/statistical mechanics: the recurrence times can be very large. 
A: as I understand this: Your first measurement will yield a result "i" with probability |c′i|2 , your following measurements will give result "i" with probability 1.
A: Note that, even without "reduction postulate",  given  any normalized initial state and any normalized final state, you may find a unitary transformation which transforms the initial state into the final state. 
Of course, it will be a one-shot unitary transformation, depending only on the initial state and the final state, so you would not be able to "control" this transformation , to "reproduce" it, to associate some predefined hamiltonian, and so on...
For instance, suppose that the initial state is $|i\rangle = \alpha |+\rangle + \beta |-\rangle$, with $|\alpha^2| + |\beta|^2=1$, and suppose that the final state (after measurement) is $|f\rangle = |+\rangle$ 
Now, it is easy to see that the unitary matrix $U(f, i) = \begin{pmatrix} \bar \alpha& \bar \beta\\ -\beta&\alpha\end{pmatrix}$ makes the job, that is $|f\rangle = U(f, i) |i\rangle$
All this is coherent with quantum mechanics, you may consider, that you have the probability $|\alpha|^2$ to have some unitary evolution wich brings you in the state $|+\rangle$, and the probability $|\beta|^2$ to have some unitary evolution wich brings you in the state $|-\rangle$
However, these unitary transformations are generally not unique, so the above example is very particular.  
This can be extended also, when considering more realistic states with interactions with measurement apparatus and/or environment. For instance, the states : 
$|1\rangle = (\alpha |+\rangle + \beta |-\rangle) \otimes |M_x\rangle$
$|2\rangle = \alpha |+\rangle \otimes |M_+\rangle + \beta |-\rangle \otimes |M_-\rangle$
$|3\rangle = |+\rangle \otimes |M_+\rangle$
(where $|M_x\rangle, |M_+\rangle, |M_-\rangle$ are normalized states of the measurement apparatus), are all normalized states, corresponding respectively to a state without measurement apparatus, a pre-measurement state (with measurement apparatus, but with no explicit measurement), and a post-measurement state.
So, you will be always able to find a unitary transformation, which transforms $|1\rangle$ into $|2\rangle$, and an other unitary transformation which transforms $|2\rangle$ into $|3\rangle$, and a unitary transformation which transforms $|1\rangle$ into $|3\rangle$
Of course, these unitary transformations will have the same extreme limitations already descrived above (one-shot transformation, depending only on the initial state and the final state, not "controllable", not reproducible , not associated to  some predefined hamiltonian)
You may have a look to consistent histories, too, for instance, here and here
