Power and current contradictions in transformer secondary
Will increasing N will increase the power?

Increase N on the secondary will increase the voltage on the secondary. If the load is purely resistive, increasing the voltage on the secondary will increase the current and power.

but the power in the primary side stays the same, no?

No. The power drawn by the primary side is equal to the losses plus the power consumed by the secondary load.

Is changing the resistor in the secondary changes the current in the primary?

Changing the resistor on the secondary side will change the current on the secondary. This will change the current on the primary.

How does the energy conversation holds true? V1 * I1 = V2 * 0 = 0 ?The second contradiction is what happens when the secondary is open circuit. We know that the power on the primary is V1 * I1 But the secondary is V2 * 0 = 0 How does the energy conversation holds true? V1 * I1 = V2 * 0 = 0 ?

If the transformer were lossless, when the secondary is open circuit, the primary would draw no power. In reality, transformers have losses, and even when the secondary is open circuit, the transformer will waste some power.