1) how do I choose C2 value? does the signal frequency and/or the op-amp output current (when driving R3 & coil) matter in the choice of the capacitor C2? How? How I don't think that if the current is several amps (just as an example) a tiny pF capacitor would work, or maybe it does?
With your example value of 100 ohms, you won't go above 20mA even with an infinite capacitor.
The amount of capacitance you need (to get full signal conduction) does depend on frequency. Higher frequency = smaller capacitor. For example, at 1kHz with a 1µF cap, you get a peak of about 8mA. But increase that to 22µF and you get 17mA.
2) does the value affect/change the phase across the inductor compared to the original signal (OP-AMP output sinewave)? If so would a small one keep it to a minimum?
3) If any value would work, any benefits in a large value (or small one for that matter)?
Yes and no. The C, R, and L (inductor) form a high-pass filter. If your drive frequency is too low, the capacitor will be high impedance. If too high, the inductor will be high impedance.
In the case of 200kHz, this would be 'too high'. You can use a 1µF or greater cap. You aren't interested in the cap's affect on the response (it should just conduct your signal and block DC) so you might as well make it bigger like 10µF or 22µF.
Near the two poles of this filter, the phase will change. I've attached a screenshot of a simulation for 22µF. Green is the current through L1, red is the voltage at OUT. If you swap R3 and L1, you can sense the current as it will be proportional to the voltage across the resistor.
For detecting metals with a coil, you are expecting the inductance of the coil to change. To sense this, the drive frequency should be near the inductor's pole so that the output amplitude changes (phase will too). The area of interest (as seen in the graph) is around 30kHz - 100kHz. This region is where the inductor's impedance is increasing and therefore the amplitude of OUT is increasing.
If you want to play around with values, I suggest you get some circuit simulation freeware such as LTspice, TINA-TI (this is what I use and prefer, it's easier to use than LT), etc. The second screenshot shows how the frequency response changes as L1 increases from 80µH to 160µH (for example).
4) I don't see the point of using R2. It was from a reference schematic. Does it serve any practical purpose in AC coupling or otherwise?
R2 pulls the right side of the cap to ground. It is intended to be a high value - 10k and above.
You don't really need it in this circuit because R3 and L1 already pull it to ground.