How to calculate the number of windings of a unknown coil?
If you can access the central part of the coil i.e. the bit that defines it as air-cored then you can calculate the number of turns. To do this you would fabricate a closed core using ferrite material and wind another coil of set number of turns around the fabricated core. It doesn't need to be a tight fit - the ferrite material should be chosen to have high permability so that magnetic lines of flux largely follow the path of the ferrite.

This allows you to couple about >95% of the flux produced by the added coil's windings through the unknown coil's windings (or vice versa). This turns it into a transformer and if you generate an AC voltage on one winding, the voltage on the other winding will be proportional to the turns ratio.

Better to choose high permeability core material to cut down on leakage flux i.e. flux that is produced by the driven coil not coupling to the measured coil. However, if you measure the coupling ratio in both directions you can more accurately calculate precise turns ratio.

You also should choose an AC frequency that is as high as possible but not so high that the core material losses play a role and produce inaccuracies. Something around 10 kHz is likely to be reasonable.

Inductance is easier - if you have a signal generator and a few capacitors you can drive the coil to resonance and, knowing F and C calculate inductance from the formula: -

F = \$\dfrac{1}{2\pi\sqrt{LC}}\$