The correct office that the daughter could apply for is 'Administrator'. An 'Executor' is only appointed when there is a will.
Apply to the court for letters of authority. If the situation is properly documented, it should not be a problem.
The executrix is responsible to distribute the assets according to the will or the laws. The consent of the beneficiaries is not required.
Executrix
That would be one of the key duties of the executrix. They have to inventory, value and liquidate the estate.
The fee is set by state law.
Yes. Your sister can file a "Declination" and the petition to have the will allowed can list you as the executor. The attorney handling the estate will know how to file the proper documents so that you can be appointed.
That is the option of the executrix. They don't have to get permission from every heir to manage the estate.
The executrix owes nothing to anyone. The estate is distributed according to the will or the law. In some cases the law may make exceptions to the will, particularly if the children are minors.
The executrix has a responsibility to keep proper records, and in England and Wales would be liable for the rest of her life for her conduct of the estate. The short answer to your question is: no.
The executrix is not personally responsible for the medical bills. The estate has to pay off the debts. As long as the estate can pay them, they have to receive and pay the bills. If the estate cannot do so, they distribute as best they can. If the court approves the distribution, the debts are ended.
There is no proscribed time frame. It can take years, even decades, to settle a complex estate.
That will depend on what the will says. In most cases, the bulk of the of estate would be expected to go to the spouse.