They have a RAT (ram air turbine) that deploys automatically under specific conditions. It’s basically a turbine providing electric and hydraulic power. It was almost certainly deployed on the accident flight. It will only power the most critical equipment, though. Possibly, that does not include the ADS-B transmitter (which broadcasts position and related data).

Even on light GA aircraft the radio will run for an hour on the starter battery. This narrative doesn't sound realistic.

Yes, lithium ones, those were the ones that kept catching fire then the 787 was just out.

I guess they're just for the time until the RAT kicks in. Or to augment it.

Ps there's also the APU, a small turbine in the tail for generating electricity.