Parallel circuit and Ohms law
If I have a parallel circuit of 2 resistances, both zero Ohm, the current would split to 2 and each current would be half. 


"Zero ohms" is an idealization. If you say you have two zero-ohm resistors in parallel, it just means your model is not accurate enough to determine how the current is split.


  If the resistor R1 would be 1 pico Ohm, and the ammeter zero Ohm, there would be no current through the resistor, and all the current would go through the ammeter?


If "zero ohms" means much much less than 1 picoohm, then yes, essentially all the current would go through the ammeter.

But real ammeters have burden resistance that's much much more than a picoohm (more like a few milliohms).


  2) Ohms law.
  R2 is 10 Ohm, current is 1A.
  voltage between point A and B is 0 volt, resistance between A and B is 0 Ohm,
  according to Ohms law, I = V / R = 0 / 0 = 0 A.


You have a false conclusion. Zero divided by zero is not zero. It is an undetermined value. Could be zero or could be infinite, depending on the situation.

To analyze this circuit, restate Ohm's law as V = I R. You know the current is 1 A due to the other circuit elements. You know the voltmeter doesn't pass current. Therefore there's 1 A passing from B to A, and because it's a perfect wire, the voltage is zero.