I am not a native speaker
About your question I think that:
1) "I don't think it will rain tomorrow" sounds better than "I think itwon't rain tomorrow", but maybe both of them can be correct.
2) I have an analogy opinion about your example with copier: "I don't think that it's working" sounds better than "I think the copier is not working", but, also, both of them can be correct.
Nickle. Not a native speaker.