1. I doubt it does, at least properly.
2. It doesn't.
3. Depends on the battery but as much as about 0.25 A.
4. If you assume a C/10 charge rate, about 15 minutes but much longer might make it explode.
I think the schematic may be wrong or it could be using a constant current LED. It certainly will not work safely as it is. As far as I can tell, D1 and R2 are there to divert some of the LED current so it dims as the voltage across the LED reduces but there is nothing to control the charge rate or cut it when the battery is fully charged.
Brian.