Is it permissible for the widow, while in her waiting period (iddah), to visit her sick mother daily from 6:00 PM until 1:00 AM, given that there is someone else to care for the mother, and would this be considered as not having left the iddah?

There is no harm in a woman observing 'iddah (waiting period) going out for a need, even if it is minor. The Prophet, peace and blessings be upon him, permitted her to go out at night to a neighbor's house for conversation and companionship, provided she returns to sleep in her own home. This is the correct view held by the majority of scholars, in contrast to the Malikis who believe she may go out during the day even if there is no need. Therefore, your aunt may visit her sick mother if she feels safe and returns to sleep in her home.