Your child and your sister's child share a common grandparent and are first cousins. Your grandchild and your sister's grandchild share a common great grandparent and are second cousins. There is a generation out of synch, so they would be second cousins once removed.
You and your uncle's great grandchild are first cousin, twice removed.
The grandchild of your great grandmother's sister is your second cousin, once removed.
Your sister would be her great aunt and she would be your sisters great niece.
If the daughter is biologically related to you, it would be your great-grandchild. If your husband had his daughter before he married you (which would make her your step-daughter), it would be your step-great-grandchild.
The grandchildren would be the parents, aunts and uncles, and first cousins once removed, of the great grandchildren.
Your grandmother's sister is your great aunt. Her son's child is her grandchild. You and your great aunt's grandchild are second cousins.
You are great great aunt to your niece's grandchild. Your children and your niece's grandchild are first cousins, twice removed.
No, there is nothing wrong with it. People are not likely to be able to figure out that there is any relationship, and if the great great uncle is from the "other side" of your grandchild's family, he is not related to you at all.
Your her great great aunt!
Your grandmother's sister is your great aunt. (Some would say grand aunt). Your great aunts' grandchild is your second cousin. Her great grandchild is your second cousin, once removed. .
Your sister's great grandson is your great great nephew. You are his great great uncle, or great great aunt.
Your grandchild through one daughter and your great grandchild through another daughter are second cousins, once removed.