How to measure distance/position accurately

You might as well combine the measurement and replication into one system. Let me start with my previous answer to your question: Idea 4

You already have linear bearings and encoders set up to measure the motion of the object. Now all you need to do is to actuate those bearings. Normally, if you were building something like a CNC milling machine, and you wanted to actuate linear motions, you'd use a ballscrew and a stepper motor.

The problem with these is that they are not back-drivable, and therefore you won't be able to move the object yourself. Two options:

1. Add force sensors (using strain gauges). Now the system can detect you pushing on the object, and it will drive the motor to create the illusion that there's no motor there at all. This can work extremely well if you do it carefully, and if the motor is fast enough. This is known as active compliance.
2. Use linear motors. These are completely back-drivable when unpowered.

The good news is that, if you can afford it, you can buy complete linear motor motion systems from companies like Aerotech or Baldor Motion. Alternatively, you can ask a small robotic engineering company to build you the whole system. Again, these systems are very accurate. They're used where 0.01mm accuracy is a common requirement, and even 0.001mm accuracy (but then you really need a temperature controlled environment).

As Dave Tweed already mentioned, the fact that the maximal separation is comparable to plates' dimensions makes this setup problematic. You may get accurate estimation of the distance while the plates are close together, but this setup won't work for the whole range.

Dave suggested that these nonlinearities may be accounted for, but I don't see how this can be achieved, satisfying the required accuracy, without very complicated calculations.

However, since you're going to use microcontroller, you may try the following trick: perform initial mapping of distances to capacitance, store this data in microcontrollers memory (assuming it is sophisticated enough) and use the stored data as a look-up-table to map the measured capacitance back to distance.

As to the required clearance, it depends on which objects may be present in the vicinity of your setup. Consider shielding it with conductive screens.