It is perfectly correct. It's the use of the English subjunctive, which we often have a workaround for, and many people have not actually learned in school (unless they studied a foreign language that uses it more) The conjugation "she live" over "she lives" is the subjunctive form.
From some comments above from a couple months ago, it SHOULD be fine in English to ditch the "that" but duo can be iffy on when it excepts stuff like that.
I think in oral conversation, you'd be a lot more likely to say "She has to live" and when I am translating the "il faut que" structure into English for things other than small Duo exercises, I'll usually make that shift. We are not so into the more removed "It is necessary that..." structure in English.
Are you from the UK? Funny that you would hear it as American. If anything, the cold formality of it makes it seem more English to me. (And yes, I know that plenty of English English is not at all cold or formal-- just thinking of Received Pronunciation and a BBC broadcaster reading a sentence like this in a formal statement to the press.)
You have probably heard it and probably use it (I am in the US also) but were not officially taught it because we learn these things intuitively, and because there are not so many forms that are different from an indicative present. You wouldn't say "It is necessary that I am in New York next week" because it sounds wrong. You'd say "that I be in New York" in that context. (One of the rare irregular subjunctive forms in English) I didn't learn that it existed until I started studying romance languages in high school, but recognized the form when presented with examples.
I got the word-picking version of this and although I've been translating all the others into sentences with English subjunctive, this one has the word 'lives' but not 'live' - I thought this was fair enough since it's so common in contemporary spoken English at least to simply replace subjunctive with present, even though it technically isn't correct - however, my answer wasn't accepted because I didn't include 'that' - this implies that with the subjunctive the 'that' isnt necessary in this semtence, but when you use present instead you need the 'that', which is clearly not the case, so it's definitely just an error, a missed possible solution, so I've reported it.
i just recognized for the first time that the subjunctive exists in english (my native tongue doesn't have that at all). is it legit to just use the simple present form (including an additional s in the third person plural) instead? i think i did that for the past 15 years and no one ever corrected me. and if it's wrong, how recognizable is it for english native speakers?