I used to love these little motors when I was teaching the basics of electromagnetism. I suppose it would be possible to set up a full mathematical model but given the number of variables involved it would make a fun (experimental) toy very boring - I say learn by doing. What's nice about this question is that it is not someone's homework - it is a genuine 'want to know.'
Some basic theory and equations for you to ponder on.
The current flowing in the coil produces a circular magnetic field around each wire (Oersted's Experiment - the corkscrew rule). The bigger the current, the greater the magnetic force it produces. Now each conductor (wire) in the coil contributes to the force. So the greater the number of turns the greater will be the turning force for a given current.
The direction of the current will control the direction of the field (Flemings Lefthand Rule)
The coil construction means that the current (I) effectively changes direction on each side (going out and coming back) producing a turning effect as it sits in the magnetic field (B) produced by the magnets. For a wire of length L sitting in a magnetic field B the Force can be calculated using F = BIL. With N turns this is simply multiplied by N.
What the formula tells us is that if we increase the magnetic field strength by using stronger magnets, or reduce the air gap distance or use a ferrous material we get more force per unit current - we can improve the motor.
Geometry is important - The length of the conductor in the field should be maximised. A circular coil has only two points at right angles to the field so you could try making rectangular coils with the long side parallel to the rotation axis and see if this has an effect.
The force produced turns the motor but does not control the speed of rotation. This is where we need to see the motor as a generator.
A conductor (wire) that moves through a magnetic field will have a voltage induced across it. (The basics of transformers etc.) This is referred to as the induced voltage or back e.m.f. or -E = BLv (Faraday's Law) where L is the length of conductor and v its velocity (a vector quantity).
We can use Fleming's Right Hand Rule (Generator) to get its direction. This is in the opposite direction to the current/voltage driving the motor.
Only the difference in the applied voltage and the back emf is actually moving the current flowing in the rotor. That's why the initial current in a motor starts high then falls off rapidly as the motor gets up to speed (increasing the back e.m.f). It also means that if you stop or stall a motor the high current that will flow is liable to burn it out.
It can be calculated from
This coil resistance depends on the length, cross sectional area (wire gauge) and resistivity of the material. (Thinner wire, higher resistance, lower current per volt)
Your tapping experiment probably improved the contact resistance with the bare wire 'brushes' or overcame some friction (stiction)due to misalignment etc. The motor was in a stall (high current, low speed). When you tapped it the speed increased, the back emf increased and the current reduced and it took less power. Been there, done that got the T Shirt.
I hope this long winded explanation gives you some insight to the wonderful world of 'simple' electric motors.