"They live here."
Translation:Они здесь живут.
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Okay I did a lot of research online and it seems they are synonyms meaning "here" (in place or time) with the following subtle differences:
тут It is more informal/colloquial. You don't use it with your friend's grandparents for example. It is also used to mark changes at a point in time. "Тут заходит дочь." And then the daughter came in.
здесь More widely accepted formal situations, but doesn't sound formal (has a neutral feeling).
http://cooljugator.com/ru/%D0%B6%D0%B8%D1%82%D1%8C this website seems super useful (I was confused on this one too).
ь is not pronounced but affects pronunciation, whereas ы is a "letter" unto itself. As with other words that use one or the other, in this case you'd just have to remember how здесь is spelled. It might be helpful to remember that it has only one syllable, but ы would add another vowel sound, making it pronounced "zdess-ih" (which is not a word).
Well, first of all "они" and "живёшь" are mismatched. "Живёшь" means "(singular) you live". It's always "живут" with "они" regardless of the word order.
As for the word order: if pronounced in a neutral tone or written without any indication of emphasis "они живут здесь" informs where they live and "они здесь живут" informs what they are doing here. In speech though it can be pronounced with a different emphasis, reversing the meaning. But that's rather complicated matter for beginners, so Duo sticks to the neutral word order.