I am 63 with 3 children (under 16) that are in the custody of my ex-spouse. Are my children automatically eligible for benefits off my record if I start collecting? Or do I need to prove a certain level of financial support (or other requirement) since they don't live with me full-time?
Biological children are deemed to be dependent on their natural parents, meaning that financial support isn't required in order for a child to potentially qualify for child's benefits on their natural parent's record. Adopted children are also deemed to be dependent on their adoptive parents. So, assuming that the children you refer to are either your natural or adopted children then they won't need to prove that you've been financially supporting them in order to qualify for child's benefits on your record when you start drawing your benefits.
However, your children wouldn't automatically start drawing benefits when you claim your benefits. An application for child's benefits would need to be filed on their behalf. Normally, the parent who has custody of the child(ren) would be the proper person to apply for their benefits.