Lets approach this (stability) question another way.
The photodiode has some capacitance. That Cdiode in conjunction with the feedback resistor, forms a delay and also a phaseshift and a pole.
1) the circuit already is inverting, thus provides 180 degrees
2) the Cdiode + Rfeedback is another 90 degrees (above some frequency)
3) the opamp provides another ...
Do we use a similar filter after the photodiode to get rid of the out-of-signal-space noise, before sampling?
Generally, we don't have a separate filter device or circuit. We'd rather connect the photodiode directly to a trans-impedance amplifier (TIA) chip to avoid losses due to impedance matching (the photodiode produces a current signal so we'd rather it ...
In wireless communication, we use a matched filter to suppress out-of-signal-space noise.
No, we do that to maximize SNR under white additive noise. That's not out-signal-space!
After this filtering we do sampling.
Ah, yes, that's what's said in many textbooks. It's not really what we do, at least for the last 25 years: it's very hard to actually ...
To answer your questions:
Gain is not really relevant, you can cascade as many amplifiers as you want. What matters is NEP, that sets when your SNR is limited by detector noise vs. shot noise. What NEP do you need?
Sure, amplify as much as your want. It does not help SNR.
Specs are not well defined yet. Can't specify if it is possible.
APDs have lower NEP, ...