Can a bank refuse to allow a person open a checking account

Banks have the right to choose not to do business with potential depositors. If they have information that indicates that a potential depositor failed to honor obligations with them or a prior bank, they are able to say no to a new relationship with you.

I am unclear about what you are saying regarding them taking a check from you. If they have funds of yours and yet won't allow you to open an account with them, they have no right to retain yoru funds.