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.
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 second cousin's daughter is your second cousin, once removed.
Your second cousin's child is your second cousin, once removed.
Your mother's second cousin is your second cousin, once removed.
The son of your second cousin is your second cousin, once removed. You are also his second cousin, once removed.
Your grandmother's first cousin is your first cousin, twice removed. Your grandmother's second cousin is your second cousin, twice removed.
Your mother's first cousin is your first cousin, once removed. Your mother's second cousin is your second cousin, once removed.
If your second cousin's mother is related to you, she is your first cousin, once removed. Remember that one of your second cousin's parents will not be related to you.
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.
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.