“My daughter has the flu so I’ll stay home from work and mother her.” Sounds fine in AmE “Mother” can also mean “give birth to”. That usage would be rare when referring to people, but ok for animals. But your example sounds odd to me (a friend wanting to mother you) “Mother” can also have a negative connotation. Talking about a married couple: She should stop mothering him. (Best) She should stop acting like his mother. (Good) She shouldn’t mother him. (Ok)

it's common enough in the UK it means to care for like a maternal mother, or as some mothers do from their child's birth to either the child's or mothers death. But it can have negative meanings like David suggested. "a friend to mother you" does seem a little odd. "a true friend to care for you" is more natural to say this https://www.merriam-webster.com/dictionary/mother yes it is a transitive verb scroll down the page in the link.

Hello! Perhaps in this context "your friend is eager to other you" just means to "take care of you" or "to fuss over you" as a mother would or as someone who feels "maternal over you", would. It means to take maternal-like care of something/someone.

English allows people to invent nouns and verbs. Often nouns become verbs. For example, "to google" is a recent verb. When invented nouns and verbs become common, they are included in dictionaries. My dictionary does include this as a verb. Perhaps you need a better dictionary. New Oxford American Dictionary mother verb [with object] look after kindly and protectively, sometimes excessively so: she felt mothered by her older sister.

Yes <and> yes. :)