Is it permissible for children to use surplus funds collected from charitable individuals, beyond what is owed for the debt, to pay off their father's debt, given that they had previously contributed to paying off his debt from their own money?

Your efforts to pay off your father's debts are a righteous deed, and whoever eases the burden of one in hardship, Allah will ease his burden. If you have enough donations to pay off the debt, it is not permissible for you to continue asking for help. Begging is only permissible for three types of people: one who has taken on a guarantee (a debt or responsibility on behalf of another), one whose wealth has been destroyed by a disaster, or one who has been afflicted by extreme poverty, attested to by three people from his community. Anything else is illicit gain. If the donations exceed the debt, then if the donors intended only to pay off the debt, the excess must be returned to them. If they intended charity for you, the excess is yours. If they intended charity for your father, the excess is his.