OpAmp feedback resistor query
You seem to be in a tricky situation. You are trying to make sense of a circuit which makes very little sense, but you haven't the experience to realize it. The things which puzzle you are indeed indications that your predecessor is better off somewhere else.

You can get equivalent performance with a single op amp with a 300 k feedback resistor. Since you are sampling at 2.5 kHz, a 500 pF feedback capacitor will produce a single pole lowpass response down 3 db at 5 kHz. The inversion produced by the second op amp can be handled by reversing the polarity of the photodiode. And, of course, the use of compensating RC at the + input (especially the capacitor value) makes no more sense on the second op amp than it did on the first.

The output filter produced by R28/R29/C38/C39 has a response down 20 dB at 200 kHz, so it's not at all clear that it does you any good. Although it may be recommended by the ADC manufacturer.

The parallel 39k resistors are only useful if you need the option of selecting a gain which can be varied by a factor of 2 by installing only 1 of the resistors. This is potentially useful, but if you're not using that option as part of system tweaking it serves no purpose.

I suspect that the purpose of using the two amp configuration is to provide limiting in case of exceptionally large optical pulses. If the TIA limits at -35 volts the attenuation produced by the second will give an output level of just about 5 volts, which sounds like a reasonable ADC input range. Since the AD8676 is a rail-to-rail op amp, you can do this for a single amp configuration simply by using +5 as the + supply voltage. +5 and -10 would work just fine.