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The people who suffered the loss, in this case the parents.
In the United States a minor child is entitled to a portion of a deceased parent's estate. In some states an adult child is entitled to a statutory portion of the estate of a parent who died intestate. Any child who is next-of-kin to someone other than their parent would be entitled to a share of an intestate estate. You would need to be more specific and check your state laws.
A proven biological child is usually entitled to a portion of her father's estate.
That depends on the laws of the State and whether the decedent left a will, but the older child would likely be entitled to a share of the estate, assuming paternity was established.
Since this is the adoption category I assume you mean not the biological grandchildren but foster kids who have never been adopted. And no, they have no right at all to anything their foster family leaves behind unless they are mentioned in the will. If you mean biological children they inherit their parents who in their turn inherit their parents, your grandparents. So unless the grandchild is mentioned in the will or the parents are deceased, the grandchild will not inherit the grandparents.
Each state has its own laws regarding this issue. Generally, a child can be disinherited by will but some states require the child to be specifically mentioned in the will or the court will assume the child was forgotten and award a share of the estate. You should consult an attorney in your area who can review the will and determine what your options are, if any.
No. Adoption removes the rights to the biological parents' estate.
Only if you adopted the child.
The estate is responsible for paying off any debts. However, in most cases if the debts exceed the value of the estate, the child is not required to pay the difference. This may not keep the collectors from trying.
Under Nebraska law you cannot completely disinherit a child. The child is entitled to an elective share of the estate regardless of what language is placed in the will to attempt to disinerit. See In Re Estate of Peterson.
You can check the laws of your state at the link in the related question below.
A foster child have no birth right after their foster parents so it's only if they are mentioned in the will.