Yes. The MobilePal wireless power bank has intelligent overcharge protection, so it is safe to keep it plugged in.
When the power bank is connected to an AC adapter, the power bank can charge the external devices in the pass-through mode. In this mode, depending on the AC adapter that you use, the charging voltage may be low when the battery of the power bank is low. If the power bank does not charge your phone, you might need to press the power button on the power bank once to switch from charging the internal battery of the power bank to charging your connected device.
In terms of battery life, the built-in battery is made of top-grade lithium-ion polymer, similar to those used in Apple iPads. When the battery is full, the power bank automatically switches to trickle charge mode to extend the life of its internal battery.