"Ониздесьживут."

Nope. Both "Они здесь живут" and "Они живут здесь" are correct.

живут is present tense conjugation of the verb жить. It means that at the present time, they are living (here) (or, it could mean they are still alive). проживут is the conjugation of the perfective verb прожить - since it's perfective, it will reflect a completed action in the future when conjugated like that, and I can't think of a good example to use it - it is much more common in the past tense. At any rate, they are not interchangeable.

The inclusion of "do" in English adds some subtle meaning to it, either to highlight the fact that yes, they live "here" or to specify that they live here and not somewhere else. In my opinion it is unnecessary since the Russian is straightforward. That's my American take on it, anyway (always add that since I know there are many things that have different meanings/usages here than in other parts of the Anglosphere.)