Assuming the diode resistance >>500 ohms at 20uA bias, your bandwidth will be determined by the 500 ohm*100pF time constant, which I think is what you are saying. You said previously you needed 500MHz bandwidth. You don't even have 5MHz bandwidth. It does solve your amplifier bandwidth problem, though.
As I said, the reason for the negative output voltage is the amplifier's bias current. In the sim below, I reduced it somewhat by eliminating the RC filter after the detector filter. I then measured the bias current and added a current source to provide that amount of current (not a good idea in practice). In addition, I duplicated the circuit, but with a behavioral op amp which has nearly zero input current (Rin=500Meg). In this, I omitted the current source, since none is needed. From the sim, you can see that the two detector outputs (k and k1) are nearly identical, but the output offsets are different. This is due to the bias current in the inverting input, and the input offset voltage (2mV) built into the model for the AD8009.