A question about differential measurement system
Consider the following differatial amplifier circuit.







simulate this circuit &ndash; Schematic created using CircuitLab

Note the input to the op-amp is a voltage offset and is not referenced directly to the system ground. As such you are pushing/pulling current to/from the +/- pins. The op-amp uses that difference to operate regardless of ground.

The key then is that both inputs must have their own common connection. Whether that be through AI ground, or completely isolated.

As for grounding. To answer your second point first, a connection to earth ground is beneficial for a number of reasons including static build up protection and EMI/EMC reasons. A single point connection is preferred such that the connection carries no current.  

Connection between AI Ground and GND on the other hand is more complicated. If they are close then they should normally be connected. If on the other hand the AI Part is on the other side of the building and or there are large currents flowing though the ground bar, then differences in the ground level can introduce unwanted effects if you connect them together. It therefor gets complicated quickly and significant amount of time and effort needs to be put in to come up with the best grounding solution for each application.

ADDITION: On re-examining your question I noticed you are using an instrumentation amplifier not a differential amplifier. That changes things since Instrumentation Amplifiers have a buffer stage on each input. For this type of circuit you may need some common reference.