Is the value of the deceased's medical bill considered a debt that must be paid after his death? And what should be done with the donated amount if the hospital accepts the appeal and waives the claim for the debt, or if it rejects the appeal, should the bill be paid from his estate?

The deceased is tormented by the debt he owes, and the torment is lifted by settling the debt. If the treating entity stipulated a full recovery and it did not occur, then they are not entitled to anything. However, if the contract was a lease (ijara) and the entity performed all the work, then they are entitled to the full wage, or a proportion of what they performed. If the treating entity accepted the transfer (hawala) to your brother's company, then they have no right to demand payment from you. But if you were the guarantor, then you are responsible. The money collected, if it was given as charity (sadaqah), then it is the property of your brother and is disbursed to his heirs or to pay off the debt. If it was for the purpose of treatment only, then it must be returned to its owners. If it is impossible to return it, then it should be given as charity on their behalf. The promise of the company manager to cover the treatment costs is not binding on him.