Most mains powered ZigBee devices are repeaters (except bulbs because folks tend to turn off the switch). See this topic:
Battery life is a function of usage and how much radio power / retransmission is needed (keeping aside the hardware design). It seems to have a longer battery life when used with a repeater.
For closing the garage door through LUM just assign a door/lock specific/individual “lock” action to close the garage when the keypad is locked.
EDIT: Regarding battery, don’t forget to set the battery type (Lithium, Alkaline, Rechargeable) in the device preferences page (gear icon) as it reports the remaining battery life based on the battery chemistry/discharge curve.