D.B.
Welcome to Mamapedia, M.!
You have a very complex situation here. If I understand you correctly, you have interference from both your own side of the family (your aunt) and your ex (your daughter's father). I'm not sure what you are saying about your cousin (your aunt's son and his wife?) - are they just reporting the rumor to you, or are they also threatening? Regardless, your own aunt is supposedly teaming up with your ex to take your daughter?
It's hard to know if these are real threats, or if they are doing this to mess with your head for some reason, perhaps to get some concessions from you, or perhaps because they are claiming you are unfit. However, even if you were unfit (which I'm of course not saying you are), your aunt would have to prove it in court, and your ex would have a lot of trouble coming forward into the spotlight if he's not here legally.
I'm not a lawyer, but it seems you have two problems: 1) a threat - real or vague - of kidnapping, and 2) custody in the event of your death or incapacitation. I don't know what your daughter's father's rights are in terms of visitation and custody. There are separate laws that relate to taking children out of this country.
Your aunt would not have your child's birth certificate or or passport, nor would she have any letter from you giving permission to take your child out of the US. (My mother took my son to Bermuda for a week, and I had to sign a notarized letter and provide his birth certificate.) Are yo sure that your ex would be stupid enough to present himself to border officials, given that he has no documentation, and also try to take a child out with no documentation?
As for custody in the event of your death, you can designate in your will whom you want to take your child. You might have a second person in mind in case your mother isn't able, but that's up to you and a discussion with a qualified family attorney. I don't know if your ex would have any rights if he has no documentation. Is his name on your daughter's birth certificate? If not, he'd have to file for custody, and that would be unlikely I would think - would he really put himself in front of a judge or magistrate if he has no legal status here?
I'd start with a family law attorney, with the understanding that he/she might have to consult with an immigration attorney or one at least familiar with immigration law. If you are close with your mother, and it sounds like you are, it might make sense for you both to visit an attorney together - one to talk/ask questions, and one to be a second set of ears as well as someone who can take notes. Anyone who has a child needs to have a valid will anyway, so that lawyer can help you do that with a "boilerplate" document where you sort of "fill in the blanks", including "custody of the minor child", and then add in any additional provisions that may be necessary. I know it's an expense, but you need this.