High gain and high modulation frequency photodetectors
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, but introduce multiplicative shot noise. Hence, if you can be shot noise limited with a standard TIA photodetector, it will perform better than the APD even though the APD has a much lower NEP. 
Probably, but you need to explain what you are using.