Is this also the way to say, "They have done the reading for today," or is that completely different?
No. "lit" as a noun only means "bed". The noun "reading" is "la lecture". "lit" is however the third person singular conjugation of the verb "lire" ("to read"), i.e. "il lit un livre" ("he reads a book/he is reading a book").
Remember, French is not like English - a noun cannot be used as a verb and vice versa.
Ex "He has the run of the place". and "We run on the weekend"
These are possible in English but not in French.