Yes, yes and yes. But the sickness must be reasonably thought to have a chance of causing death, on its own. No sniffles. No depression. Etc. This is the teaching of all the magisterial documents that address it.
If an otherwise healthy young man develops a bit of flu, he is not in reasonable danger of death and is therefore not a valid subject of this sacrament. If a 90 year old man contracts an identical condition, the same may not be true.
The illness must be a lethal threat to the subject of the illness, not merely lethally threatening in general.
This follows from the primary purpose of this sacrament, which is the spiritual preparation for death. Such grace is guaranteed (though it still requires the proper disposition, even if only implicit, to make this grace effective in the soul). Physical healing is an occasional side effect.
I hope this helps to clear the difficulty. Consult the article linked to above for more details.
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