Do batteries on new devices start discharging even when plugged in to the adapter? On my old phone, HTC Hero, when it was connected to the charging adapter and the battery level reached 100%, it never went down when plugged in, like the battery wasn't used at all and the phone was running from the charger instead.

On my new phone, HTC Evo 3D, I left it plugged in into the charging adapter in the evening and when I woke up in the morning, the charging LED was green (implying the charging was complete) and the battery level was 98%, which seemed like the phone "discarded" the charging adapter after the charging was complete and started running from the battery, even though it could have gone from the charger.

Is this normal for all new devices, or at least for HTC Evo 3D?
The device does not run on the charger. It will always run on the battery, unless the battery is not present. 

The discharge you see can be from just applications on the device running in the background. The charger stops charging the battery when it reaches full status. Note that this may not be 100%, even on a brand new battery. My battery doesn't charge past 96% any longer. Once it reaches that, it reports full and the device stops charging the device.

The device stops charging once the battery reports full so it doesn't over charge and damage the battery. This is why you probably see that the battery is at 98% after leaving it on the charger for a while. It does charge full (any probably to 100%, or close to it) but then the device stops charging once it is full. It wont just start charging the battery again when it drops below full, because this can also damage the battery if it was constantly charging/not charging the battery.