What is the ruling on disposing of a sum of money that exceeded expectations and the will of the deceased: should it be given entirely as a perpetual charity (Sadaqah Jariyah), or should the remainder be divided as inheritance? And is it obligatory to inform the heirs in both cases?

If the father specified the amount as 6000, then this is his will. However, if he meant the full value of the document (19000), then his will is the entire amount, because a bequest of an unknown amount is valid by consensus. If there is doubt about his intention, the default is that the will is not binding except for the confirmed amount. What exceeds the bequest is distributed among the heirs like the rest of the estate. It should be noted that a will is not enforceable for more than one-third of the estate unless the heirs approve.