Log Detector Output

With no signal at the input, the AD8309 will should have a dc offset of around 400 mV. You can see this in Figures 7, 8 and 9 in the datasheet. 

I would not call the 1V you are seeing, a dc offset. What I suspect is going on is that the part is detecting and measuring noise or an interference signal. A 1V output corresponds to around -45 dBm of input signal which is around 1.3mV of rms noise. To mitigate this you need to eliminate the noise or shield the part. I would experiment with putting  power supply decoupling capacitors on the individual power supply pins (in case RF is getting in through the power supply pins). Another thing you can try is to turn off you main RF signal source. If the 1V is still present, that means that there is interference coming from somewhere else.  

I note that you have the limiter disabled. This good practice if you are targeting maximum sensitivity from the circuit. When the limiter is enabled, the large limiter output signal can leak through the board, back to the input and degrade sensitivity. But you have the limiter disabled so this mechanism is not at play here. 

Regarding your second question, all output voltages dc voltages, measured relative to ground.