Photodiode circuit Op Amp and amplification failure
I think I can help you out with the first part of your question, if you see the I-V characteristics of the photodiode, you will find that the current increases linearly for some time, then it becomes non-linear and saturates as you approach the bias voltage, and as you are getting the voltage across the 10k resistor,the maximum voltage that you can obtain is I(Sat)*10k.

As for the Op-amp circuit, your photodiode will be acting like a current source,so I think you should try it by using a resistor in series with the photodiode, so that you can give an equivalent input to the opamp.(Not sure about this part)