li-ion battery charging
You can do everything you're looking to do with a single device. YOu're looking for a feature called Power path that will do two things:

1) Charge the battery independantly from power the system
2) When charger input isn't connected, powers the system from the charger

For your issues:


Detection of charge finalized depends on setting the taper current. Typically this is 10% of the max current during full charge
Same as above. Charge continues until about 10% of full charging current


This list should get you started.