Why is no phase information lost when a received signal goes through a mixer?
The receiver's mixer introduces a phase-shift to the in-phase and quadrature components. In very general terms, assume we have a signal $$a(t)e^{j(2\pi f_0 t + \phi(t))},$$ which is mixed with a local oscillator of frequency $f_{lo}$ and phase $\delta$: $$e^{j(2\pi f_{lo}t + \delta)}.$$ The output signal is 

\begin{align}
a(t)e^{j(2\pi f_0 t + \phi(t))}e^{j(2\pi f_{lo}t + \delta)}
&amp;= a(t)e^{j(2\pi (f_0+f_{lo})t + \phi(t) + \delta)} \\
&amp;= a(t)e^{j(2\pi (f_0+f_{lo})t + \phi(t))} e^{j\delta}.
\end{align}

In other words, even though the frequency has been shifted and a phase-shift equal to $\delta$ has been introduced, $a(t)$ and $\phi(t)$ are still present in the signal and that means that all the information carried by the original signal is still there.

Usually the phase-shift is corrected in the digital back-end in the receiver, or you can use a differential encoding (such as DBSK), in which the phase-shift becomes irrelevant.