Amplifier Specifications for Lock-in Detection
Will some of these things (such as offset drift) be eliminated
  entirely by AC coupling?


Yes, coupling eliminates low frequency noise

You don't need to use an AD630, a circuit like this will do, then you can choose a suitable amplifier for your application with the right bandwith and noise. 

Source: http://www.analog.com/en/analog-dialogue/articles/synchronous-detectors-facilitate-precision.html

Another way to do the lock in amp is like this:







simulate this circuit &ndash; Schematic created using CircuitLab


  Which amplifier parameters should be optimized in this case where we
  need to minimize introducing AM noise on a high-frequency narrow-band
  signal?


The amplifier parameters will be determined by how much SNR you need for your application. The bandwidth of the amplifier will be determined by the lock in frequency (it will need to go faster and much faster than the next cycle). Offset is usually not a problem since it shifts the DC and the ADC by a few bits.