How to amplfy µA from a fast photodiode with a bandwith of 10GHz
Let's assume I have a rectangular optical pulse with a peak power of 10µW(@1550nm) and a duration of 1ns. What would be a solution to turn this into a proper voltage signal of around 150mV when I loosen the bandwidth requirement ?


10µW(@1550nm) for 1ns is about 78000 photons.  Assuming 75% QE, you'd get about 60,000 electrons from a photodiode.  You want to turn those 60,000 electrons into 150mV over 1 ns. V=I*R, so 150mV = (60000*1.6E-19/1E-9)*R.  Solve for R and you need a transimpedance gain of 15625 ohms.  Since you are assuming your "faint laser pulses" are actually fairly intense, you don't need a huge amount of gain.  

However, finding a photodiode that can meet those bandwidth requirements is going to be tough, so I think you will in fact end up relaxing your bandwidth spec a lot.  As far as commerical products go, you can buy ~1-2 GHz detectors with that transimpedance gain easily enough.