The ground of the circuit containing DUT (GND1), battery and test resistor will not be common with the ground of the INA828 (GND2) and subsequent ADC.
The whole measurement scheme depends on knowing how the two disparate ground systems work ifinteract (if you are to rely on a measurement that involves some form ofa galvanic connection to the measurement amplifier). For instance, if the two grounds wobble with respect to each other at 10 volts p-p at some frequency then it's extremely likely that your measurement system will fail due to excessive common mode interference exceeding the InAmp supply rails.
Of course, your two disparate grounds could have 1,000 volts between them and this is a real problem. So, unless you can resolve this, all four of your questions are invalidated.
If this cannot be resolved then I suggest you use an isolated measurement method. This might bemean battery powering a "front-end" circuit from the battery that locally monitors that current through the resistor and transmits that information optically (or via radio) to the main measurement circuit powered remotely. This is a real option.