I'm not the expert but I think you'd say "Sie glauben mir erneut". Notice that the order of the pronoun and verb is inversed from the sentence Duolingo proposed which makes it a normal statement, whereas the original one conveys the imperative form. Notice also that, though both "Sie"'s are written with a capital letter, mine is only because it is in the begining of the sentence, whilst Duo's is because it is the formal "You"! So it can never mean "they" in this case! :)
I'm almost sure my explanation makes it clear but if someone notices any mistake of missing information please say it.
Well, you wouldn't (or shouldn't) say that in English (US at least). But something to consider is that you have an implied "I am/feel" before the 'renewed'. This sounds horribly awkward to me to be left implied in English. In German there may be a different word for the sentiment of "renewed", but I know that the word endings should be the same as if the implied part was actually included (rather than implied). Hopefully that makes sense and is at least mildly helpful.