"I have no worries" is pretty typical. ("I've no worries" wouldn't be, but Duolingo automatically accepts "I've" anywhere it sees "I have.")
I agree that it's more common to say "I'm not worried," but that would be "no estoy preocupada" or "no estoy preocupado" in Spanish. (The idea is the same, for the most part, but the grammatical construction is different, in both languages.)
Not necessarily. The word itself gives a clue: pre-occupy - already taking up the space. The OED defines "preoccupy" as: "(of a matter or subject) dominate or engross the mind of (someone) to the exclusion of other thoughts." It's not just a pastime, but basically anything that takes up your mindspace. It has about the same range of meanings in Spanish.