"He does not live here."
Translation:Он здесь не живёт.
I'm also confused why Google writes живет instead of живёт. My grammar book says that for е-conjugate verbs, the 3rd person singular present time should use -ёт when the stress is at the end of the word in the 1st person singular. For жить the first person singular is живу́ so 3rd person should be живёт.
It's probably because Russians don't care much about the umlaut (the two dots over the ё), they write both е and ё as е. More here: https://forum.duolingo.com/comment/13114415/When-and-why-can-the-umlaut-in-%D1%91-be-omitted Yes, it makes studying Russian a little more complicated, but once you advance and memorise the words and construction you eventually end up getting used to it.
Maybe MichaelDir16's comment above might help you. I'll copy it so you don't need to search for it:
"Placing zdes [здесь] last is correct but emphasizes the importance of HERE. As in "Mom, he (the man in the courtyard ) doesn't live HERE!"
So, "живёт" on the last one to express that the thing that matters is that he doesn't live here. If I understood the rules correctly, it's in the given context always the last word that determines the emphasis of the whole sentence.