Electronic – Should a capacitor be considered functional if it’s capacitance reading is correct

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.

The OL is supposed to tell you that you have overloaded the multimeter in some way. If you look at the datasheet or the specifications of the multimeter you linked, you will notice that it has a capacitance range of only up to 100µF. So the capacitors you are trying to measure are too big (if they are good).

Another overload condition is a short circuit of the leads, it can be imagined as a infinitely large capacitor.

A note on the autoranging with overload conditions: it may take more than 3 seconds to get to the right reading as the multimeter tries the actual measuring range, detects a too big value, switches to the next, detects an overload, until it is at the largest range. I just tried this with a top range Gossen Metrawatt Energy and it took roughly 5 seconds before I got a reading on a 22µF capacitor. Before it would display OL. My HP/Agilent/Keysight 34410A takes around 8 seconds to get the reading, but the display freezes before displaying something.

When you are measuring capacitance with normal leads you might want to hold them as steady as possible and not closely parallel together. Another way is to tape them together and use the Zeroing function of the meter to cancel the lead capacitance out. Additionally you should try to remove your hands while measuring as your body will have a significant effect on the measurement.

A defective capacitor might end up at 30nF, I've taken apart the power board of my LCD monitor and there were some really bad caps, also specified for several hundred microfarad and now in the nanofarad range. Another good indicator for a defective capacitor is the increased ESR (equivalent series resistance), but only specialized LCR meters will give you that value.