1. wired is only for emergencies when the battery is dead. All ANC headphones, as far as I know, rely heavily on DSP as they are designed mainly for the ANC function. So it is normal that anything that works when switched off will not sound good.
2 But there are others who think they sound better with a cable.
3. personally, I only use the headphones via Bluetooth and they do their job perfectly. But I'm not that much of an audiophile!