How to obtain a high precision mV signal within a specific interval using a variable resistor?
You have to keep the inputs within the input common mode range. 

Typically with an instrumentation amplifier that depends on the gain as well since there are internal nodes that can saturate for some designs, however in this case the manufacturer lists the range for G=1 as +1.9V to 12V-1.4V = 10.6V. Obviously your inputs are both within a couple hundred mV of 12V so this won't work. 

It will be easier if you can ground one side of the low resistances and use bipolar supplies such as +12/-5V. 

As to an instrumentation amplifier having an advantage, that depends on whether you expect significant common mode voltages to be present and if said voltages are varying. In a typical Pt100 RTD signal conditioner (which this sounds like it is, maybe I'm guessing wrong) that's usually not true since the reference half of the bridge is local to the amplifier. A simple non-inverting op-amp amplifier would be fine in that case.  

Noise from the circuit can be minimized by using low resistances (reducing the Johnson-Nyquist noise) and using a low noise op-amp or in-amp. Noise from external sources is typically attenuated by using a filter in front of the amplifier input, which might be a simple RC low-pass or something more sophisticated. In most of the situations I run into (but not all) you can almost ignore the internal circuit noise since external noise is much more significant.