No, you can't charge the 928 without it being on (same for all Windows Phones). As for charging after it shut down with critical low battery, I haven't tried that, so I don't know if it turns on once it gets enough juice or not. I would guess that it would.
I think I remember this happening once right after I got it. Charged while off for a few minutes and when I checked later it was on.
I've never used that before for it turns down background tasks which I need in daytime. And charger would be connected during the night.
BTW,there isn't a way that 928 be charged with power off, is it?Unless that short period of power-off-charging after it shut itself down with critical low battery.