Discharge and Recharge of a battery
Under which value it is not recommendend to discharge a Li-Ion battery?


It depends heavily on the particular battery chemistry, as some Li-ion cells are capable of very rapid (>1C) discharge and others need to be discharged much more slowly. The particular charge/discharge rates should be documented in the datasheets for your particular cells.


  How fully discharging battery affects a battery life?


Fully discharging a Li-ion cell is generally detrimental to the life of the cell as it leads to formation of irreversible chemical byproducts (typically oxides of Lithium) and lowers the overall capacity of the cell. As you discharge the battery, Lithium ions intercalate back into the cathode structure, lowering the potential of the cell. Above a certain stoichiometric ratio of Lithium to cathode material, oxides of lithium and other byproducts form. These reactions are often irreversible, leading to precipitation of inactive material and blocking active surface area from participating in charge/discharge reactions.


  What happens if we leave a battery on charger, even battery is fully charged?


Generally nothing. After the voltage of a lithium ion cell reaches a set threshold (dictated by the chemistry of the cell and construction of the battery) the charging circuit switches to constant voltage mode and holds the potential at a set level until the current decays to a very low level. As lithium ions intercalate into the anode (generally graphite or graphitic carbon structures) the graphite saturates (approaching a Li:C ratio of 1:6) and the net flow of current decays. At this point, the charger stops applying a voltage and stands by waiting for the voltage to drop below a certain point and topping up the cell periodically. 

If you were to continue applying current during charge, or your charging circuit didn't switch over to a constant voltage mode, then the potential would rise to a point that would lead to over-intercalation and delamination of the graphite, breakdown of the electrolyte, collapse of the metal-oxiand eventually complete breakdown of your cell. This failure is often violent as the electrolyte is flammable, and should serve to highlight the importance of proper charging procedures when working with Li-ion cells. A good summary is given by Digikey in their Designers Guide to Lithium Battery Charging