1. To what level (the equivalent of a full charge or for cell equalization?), and; 2. Are you asking about your HV traction battery? iPhone ?
If it's an old Hyrbid battery it could help it live longer... If it's an old 12v battery it might help, but not as much as with the Hybrid...
For the HV pack, IF you use a peak detection charger and do each module individually, most likely YES (or use a commercially available pack charger/discharger). For the 12v, doubtful.
Well, it won't die as fast as it would if it were being constantly drained, it will allow the ice to run less, because it's fully charged, of course letting it sit charging after it's fully charged isn't good for it, but a timed trickle charge or an intelligent trickle charger would avoid that provlem. Last longer?? Tough one. Better gas mileage, yes. The prime charges the 12v as it's charging the traction battery.
Sure would be nice to redesign the bus bars with a mechanical switch so without removing the pack you could go between charging whole pack in a series vs. individual module testing, diagnosing and reconditioning. But my brain won't let me even think about what a sketchy hassle fabricating that would be.