This is the answer to the question. Give enough detail to adequately answer the question, but consider using multiple questions so that each answer is concise.
The Sacrament of Anointing of the Sick is administered, by a bishop or priest, to bring spiritual and even physical strength during an illness. This sacrament, unlike some of our other sacraments, can be administered more than once, especially if a person has a chronic illness.
It is important to remember that God does not always heal the physical infirmities that afflict us when we recieve this Sacrament. Sometimes this might happen, but it is not guaranteed that once we recieve this sacrament we will be healed from our ailments. It is more of a God-given grace to help us deal with what is in store for us and to help us prepare for that journey.
A good or reasonably sure judgment is sufficient for deciding on the seriousness of an illness. For example: A sick person may be anointed before surgery, whether or not a serious illness is the reason for the surgery.
Elderly people may be anointed if they have become notably weakened even though no serious illness is present. Sick children may be anointed if they have sufficient use of reason to be strengthened by this Sacrament. And if you are still unsure, when in doubt, ask Father!
No. The Catechism says, "The anointing of the sick is not a Sacrament for those only who are at the point of death. Hence, as soon as anyone of the faithful begins to be in danger of death from sickness or old age, the fitting time for him to receive this sacrament has certainly already arrived" (CCC 1514).
Last Rites refer to three sacraments administered to a person who is approaching death: Penance, Anointing of the Sick and Viaticum (Holy Communion given to a dying person). Each of the sacraments is performed to free the soul from every one of their sins in preparation for the afterlife.
When the Anointing of the Sick is administered to a person who is likely dying, the Apostolic Blessing (or Apostolic Pardon) is given along with the anointing. It is an indulgence given in situations of danger of death, usually after the absolution of the Sacrament of Penance. The focus is on the remission of temporal punishment due to sin.
Since Last Rites involves sacramental absolution of sins, only a priest or bishop can administer this ritual.
All sacraments are celebrated and administered only with/to the living. As such, the Anointing of the Sick will not be administered to one who has already passed. The same is true for Last Rites.