Terminology confusion with sensitivity used with gain
I would actually call the \$x\$ V/N number the sensitivity.

You can add amplification to make that 10 \$x\$ V/N but that does not change the sensitivity of the sensor itself. The amplification only increases the output level.

What then defines the lowest value we can measure?

If we would make the applied force lower and lower at some point we would not be able to measure a sensible number anymore due to noise. This can be thermal noise or induced noise (or radiated signals) emitted by surrounding equipment. The equipment we can remove or shield, the thermal noise we cannot avoid.

Since noise is a random signal it can be "averaged out". A noise signal will have an average value of zero (actually it is a limit function converging to zero) so the longer you measure, the more accurate you measure and the more resolution you get.

Then obviously you can measure slowly changing forces (more time to average) more accurately than quickly changing forces (less time to average).

So: resolution depends on how fast you want to measure.

An additional amplifier would amplify both noise and the signal (voltage) you actually want to measure so an amplifier by itself does not increase the resolution at all.