Matching an amplifier to a video transmitter
Q1: How do I transform the input and output reflection coefficients of this amplifier into impedances?


This question is fairly straightforward, however it depends on the reference impedance of the amplifier in question. The input impedance can be calculated from the input reflection coefficient as follows:

$$
Z_{in} = Z_0 \frac{1 + \Gamma}{1 - \Gamma} = 50\frac{1 + 0.071 e^{-j113.2(\pi/180)}}{1-0.071e^{-j113.2(\pi/180)}} \approx 46.9 - j6.2~\Omega.
$$

Similarly, \$Z_{out}\$ can be calculated as:
$$
Z_{out} = Z_0 \frac{1 + \Gamma}{1 - \Gamma} = 50\frac{1 + 0.039 e^{-j134.54(\pi/180)}}{1-0.039e^{-j134.54(\pi/180)}} \approx 47.3 - j2.6~\Omega.
$$

Both of these calculations assume a reference system impedance of \$Z_0 = 50~\Omega\$.


  Q2: Once the equivalent impedances have been found, how can an LC network be formed to achieve the required impedance matching to minimize reflections?


Since the input and output impedances of your amplifier are quite nicely matched on the package I don't think you should really concern yourself with impedance matching them. This is for two reasons:

1: At the frequency you're working at (850 MHz) most lumped components will at or beyond their self-resonant frequency (SRF), so impedance matching using lumped elements is pretty much out of the question.

2: Transmission-line matching networks at this frequency will be quite large unless you're going to use a high \$\epsilon_r\$ substrate like alumina (\$\epsilon_r = 9.6\$).