Battery charger disconnection
When the battery is full and the charger's on, the charger provides the power, probably the main reason the charger doesn't get hot is that it's not working as had. 

For example, my laptop charger is 65W and my battery monitoring tools tell me that it usually pumps up to 35W directly into the battery, at 35+ watts, the charger gets warm. However, the laptop it's self needs only 6W when running normally, the charger cools down as it doesn't need to work anywhere near as hard to run just the laptop. The power the laptop is drawing is coming directly from the charger and is bypassing the battery altogether. 

As a side note, the life cycle of any rechargeable battery (not just Lithium) is heavily dependent on how much it is discharged each cycle. A battery that has a 1000 cycle life at 80% DoD (depth of discharge) may last for 2000 cycles at 50% DoD or tens of thousands at a few % DoD, the more the battery is discharged, the shorter the life. It's not a hard and fast rule though, just because it says it has a 1000 cycle life doesn't mean it'll die at 1001 cycles. It's a gradual loss right from day 1, e.g. losing 0.01% per cycle would leave you with ~90% of your original capacity after 1000 cycles (0.9999^1000 = 0.904 or ~90%)