Yes, it counts as babysitting - a favour.
Your niece and your child are first cousins. Your niece's child and your child are first cousins, once removed. The children of your niece's child and of your child are second cousins to each other.
Your cousins child is your first cousin once removed.
Because that is the way it works. Your cousin's child is your cousin once removed because you are separated by one generation while you and your cousin are of the same generation. Your cousin's child is not your neice or nephew because that relationship is reserved for the children of your brothers and sisters.
Your child and your first cousin's child are second cousins. Your child and your second cousin's child are third cousins to each other.
Your child and your niece are first cousins to each other. Your child and your niece's child are first cousins, once removed.
The father and mother of a child being first cousins will not impact a paternity test. On average, first cousins share 1/16th of their total DNA.
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They would be third cousins, I don't believe that's illegal in any state, only first cousins, and in some states not even that. Being that distant of relation, there is no worry of their children having birth defects. They would be third cousins and I don't believe any state outlaws this. Some states do not even outlaw first cousin marriages. Being so distantly related, there is no worry of their children having defects of any kind.
Your child and your nephew are first cousins. Your child and your great nephew are first cousins, once removed.
Your child and your first cousin's child are second cousins. A full generation is another set of cousins. For example, the second cousins' children will be third cousins, etc. Now here is the tricky part. You are a first cousin once removed to your cousin's child and your cousin is a first cousin once removed from your child.
Your first cousin's child and your child are second cosins. Your first cousin's grandshild and your child are second cousins, once removed. Your second cousin's child and your child are third cousins. Your second cousin's grandchild and your child are third cousins, once removed.
First cousins once removed. Your child and your nephew are first cousins and there is one generation difference.