How does this pulse amplifier work?
Here is what I get in simulation for this circuit, given a 10mV input pulse of the shape used in the Bob Parker meter- 10usec pulses at 2kHz. 



Gain is about 18 with the trimpot centered and the midrange beta transistors used (BCXXB). Range is from about 13 to 27. A better choice for R16 would be 150\$\Omega\$ if the gain is supposed to be exactly 20. 

Temperature sensitivity of the bias is fairly large- 347mV at 0°C, 461mV at 25°C and 584mV at 50°C. Gain varies from 17.7 to 21.0 over that temperature range (with nominal gain at 19.8 at 25°C).  

There are some similar examples here - see figures 39 and 40. AC coupled amplifier with feedback using complimentary transistors. Note that the bias is optimized for low duty cycle positive-going pulses (idles close to the negative rail). 

As to how it works- R11/C9 filter noise from the 5V supply and give about 4.8V for the amplifier. R12/D5/D6 clamp then input voltage to the rails. C7 provides AC coupling. R13/R14 set the DC bias voltage at the base of Q7 to about 3.3V. 

You should be able to analyze the DC bias point analytically in a fairly straightforward manner (ignore C8/R16/VR2) and the AC gain by using the hybrid-pi models for the transistors. It's less closed-loop gain than the ratio of resistors would suggest because the open-loop AC voltage gain of the pair is only a few hundred. 

By the way, although the limited accuracy and stability of the amplifier might suggest replacing the transistors with a op-amp, you will find that the rise time of a couple hundred ns is very hard to achieve with a low-cost op-amp. This circuit is well optimized for the purpose by a very competent designer.