The common name given to the Sisters of Charity is the "Grey Nuns."
Nuns
Domenican sisters wear an all white habit, as do Cisterians nuns, and I believe Carthusian nuns.
According to the Center for Applied Research in the Apostolate at Georgetown University, in 2012 there were 54,018 religious sisters in the U.S. Strictly speaking, only those in monastic communities are nuns, though this term is popularly used for all women religious (monastics and mendicants), members of institutes of consecrated life, and of societies of apostolic life.
Nuns are solemnly professed religious sisters who live in a monastery, usually called an Abbey. Please note that Nuns are NOT Sisters. Sisters in the Catholic Church are religious women who take perpetual or simple promises, and usually live in a convent, teaching or nursing.
The Missionaries of Charity. However, they were sisters and not nuns. Nuns pretty much confine their lives to a monastery and rarely go out into the world. Sisters live in convents and work in the world.
Sisters of Charity
Sisters and nuns
Girls who were training to be nuns would be postulants when they start, and then novices. The Catholic Orders do not allow boys to be nuns or sisters.
Your question should probably be, "Do Sisters work in asylums?" as nuns are enclosed in a monastery and have no outside work.
The common name given to the Sisters of Charity was the "Grey Nuns".
No, Lucy was not a nun. Orders of religious sisters and nuns did not yet exist at the time.