Hi Andy,
Here's the answers to your questions:
1) You cannot fundamentally increase the accuracy at 0, 180 and -180 degrees of phase. The accuracy is what the accuracy is, and is inherent in the XOR style phase detector.
2) As I mentioned in this post: Is AD8302 a lock-in amplifier?, it best not to think of it as a "frequency multiplier/lock-in amp" phase detector.
3) There is no internal phase shift in the AD8302, 90 degrees or otherwise. The limited outputs of the log amps, which drive the phase detector, have the same relative phase as applied at the inputs of the AD8302.
4) The output stage(s), as seen in Figure 3 of page 16, has a 900 mV offset. This is put in there so that, given the default slope of 10mV/degree, the phase detector output will report the full +/- 180 degrees of possible phase information.
5) Using a 90 degree phase shifter is the easiest way of working around the error at 0 and +/-180 degrees. Re: AD8302 360 degree phase measurement is the usually solution we've given out in the past to achieve a 0 to 360 degree phase detection scheme.
Hope this helps,
Joel