No, your father's first cousin is the parent of your second cousin. Your second cousin is your father's first cousin, once removed.
Your mother's first cousin is your first cousin, once removed. Your mother's second cousin is your second cousin, once removed.
Your first cousin's child is your first cousin's, once removed. Your second cousin's child is your second cousin, once removed. Your third cousin's child is your third cousin, once removed.
Your mother's second cousin is your second cousin, once removed.
Your second cousin's daughter is your second cousin, once removed.
Your second cousin's child is your second cousin, once removed.
Your father's first cousin, once removed, is your 2nd cousin. Your father's second cousin, once removed, is your third cousin.
Yes, all cousins, of whatever degree, are really related to you. Every cousin is a person who shares some ancestor with you. A second cousin once removed is more closely related to you than a second cousin twice removed and less closely related than a second cousin, or a first cousin.
You and your mother's second cousin are second cousins, once removed.
Your second cousin and your child are second cousins, once removed.
Your second cousin and your child are second cousins, once removed.
Of course, your first cousin's child is related to you; you have ancestors who are also ancestors of your cousin's child. Your first cousin's child is your first cousin, once removed. Your second cousin's child is your second cousin, once removed.