Is it true that a sin is incurred by daughters, and that their not marrying is a trial for their father, due to his severing ties with his kin?

The delay in your marriage is not necessarily a punishment from Allah. It is obligatory upon your father and upon you to maintain ties of kinship. The injustice of relatives does not justify cutting ties with them, unless there is harm. The one who maintains ties is not the one who reciprocates, but rather the one who connects when his kin cuts him off. A Muslim should pardon and be good to those among his relatives who mistreat him, for he has a supporter from Allah. Therefore, you must maintain ties with your kin and advise your father to do so, and strive for reconciliation between them, for maintaining kinship extends one's provision.