It's the same in English: you can say "Tomorrow she will be busy", but that's not the original sentence. Using present rather than future tense when talking about future events conveys their expected or planned nature (in both languages).

Yes, if you visit freetranslation.com, type "he is busy", translate it to Russian, it says, "Он занят." Use this trick on your own to research for yourself any time you need to confirm your suspicion..

That really depends on what you are trying to say. In Russian, the last word is typically the one introducing new information. Hence "Она завтра занята" is a statement about her schedule tomorrow, while "Она занята завтра" stresses "завтра" and likely implies that she is free on other days.