Was Helmut Schmidt an adopted child?

If you live in the US (and without knowing the state) generally speaking, adoptive or biological is irrelevant. If there's a will and the child is not named, the child gets nothing, unless they can...

Yes. You must terminate your parental rights in order for a legal adoption to take place. All your obligations and rights to the child are terminated at that time.

Your cousin's child would be your first cousin once removed. So, your cousin's adopted child would be your adopted first cousin once removed?

A legally adopted child has all the rights of a biological child for purposes of inheritance under the state laws of inheritance and intestacy.

Not sure what your question is but that happen.