I'll answer very quickly.
one day: it's equal? Usually not. To express more than one day (off) we use the numbers (two days off, three days off...) but for just one I would prefer the way I've always used. Obviously I think you know what it means (not "a day out"... but "a day home").
I'll pay you a hundred for a week. Wrong. I know in Italian we are used to saying like this, but in English you must compulsorily use that word in that way. An option could be: per week.
Is "time" not a abstract noun? Why insert "the" in this? What does the abstract noun have to do with the use of the? Look at the Italian translation. Do you say dimmi ora or dimmi l'ora?
_ darkness doesn't worry _ cats; _ cats can see in _ dark
Here I confess, I don't understand your book's rule. Maybe it's right, but I would rephrase it with:
when your're speaking about general objects, things, people or animals you needn't put an article. When you are speaking about a precise, punctual object, thing, person or animal you must use the article. So let's re-watch the sentence:
darkness(in general, not a specific one) doesn't worry _ cats (cats in general, not your cat, for example); _ cats (same) can see in (here the same, even if you can see some speakers that put an article and in this case it is not completely wrong) dark.
I'll give you some extra examples.
Why don't you call the postman who came here yesterday? = that one, that precise one.
Postmen use to get around with their motorcycles. = postmen in general, not a precise one.