Actually, it can't be the sketch, because it worked best with a single led.
I've looked at it hundreds of times, but can't find the error. I also checked the MOSFET, it works fine. Maybe someone can help me.
Precisely as you would expect if you connect 12 V to the "5V" pin.
Did it not occur to you that connecting 12 V to a 5 V pin was not a sensible idea?
Actually, it turns out that connecting 12 V to "Vin" is not such a good idea either. The Nano actually operates on 5 V and you should power it with 5 V, either by a USB connection or to the "5V" pin and ground.
No, because you have destroyed the microcontroller chip as well, just not as spectacularly. This is not in any way in doubt.