What is the ruling on shortening the prayer when leading family members, and is it permissible for the one praying to pray alone and for them to pray alone when they are not mindful of the prayer time?

Congregational prayer is obligatory for adult men as long as they are able to perform it, so prayer is not valid if performed individually. You can encourage your family to pray early, then you can pray with them as a follower (ma'mum), or you can wait for them to pray with them as a follower (ma'mum). As for the anxiety about leading the prayer (Imamah), there is no need for it; for lightening the prayer is a response to the command of the Prophet, peace and blessings be upon him, and by doing so, you are obedient to Allah. Pray as a follower (ma'mum) if leading the prayer is difficult for you, or pray as an Imam and strive against your anxiety, seeking help from Allah and remembering that you are fulfilling the command of the Prophet, peace and blessings be upon him, by lightening the prayer.