Technically, yes. To be specific, you are second cousins.
If a cousin has a baby, the baby is a cousin one further time removed to all other cousins of the parent. If the parent has a second cousin, the baby is a second cousin, once removed, to that person. If the parent has a first cousin, once removed, the baby is a first cousin, twice removed, to that person.
second cousin
Your cousin's baby would be your second cousin
Yes, the baby would be your cousin one more time removed from your relation to the cousin. E.G - if your second cousin has a baby, the baby will be your second cousin once removed and if you have children, they will be third cousins of the baby.
your aunts baby is your cousin but unless your aunt married her the man that fathered her baby and the other bby then the other baby is not you cousin. If your aunt and the man are married then the baby he had with another woman is your cousin in law.
Your 1st cousin's baby is your 1st cousin once removed.
Your her baby's cousin
No, you would only be an aunt if a sister had a baby. If your cousin had a baby then the baby would be your second cousin.
the baby is your second cousin
The child of your First Cousin, and you are First Cousin's, Once Removed.
Your aunt's baby is your cousin.
You are the baby's second cousin.