Photodiode - Modes of Operation, Currents and Voltages
Many photodiodes, particularly silicon-based ones, will not emit signifcant light, even when driven in quadrant I.
Operation in quadrant IV (or along the I=0 axis) is normally called photovoltaic mode, not "solar cell". 
The distinction you're making between "active" and "passive" elements is not the way these terms are normally defined. For example, a transistor is considered an active element, but it doesn't deliver power to the circuit. In categorizing the photodiode operating modes, I'd just call these power-generating and power-absorbing operation.



  How do the generated electron-hole-pairs in the depletion region behave for different modes of operation?


In either case, the electric field in the depletion region has the same sign, so the generated carriers flow the same way (electrons toward the cathode, holes toward the anode).


  Is this true for both forward and reverse bias voltage, whether it is internal or external? 


Yes. Forward biasing just reduces the width of the depletion region and reduces the potential barrier, so that thermal energy is enough to get some carriers across the depletion region. The depletion region field is never reversed (practically, without destroying the device).


  What does happen next?


Once the photo-generated carriers are swept out of the depletion region they form an excess carrier concentration in the bulk region, creating a diffusion current.