Translation:You will receive an answer next week.
Wait why is this the locative? I thought it would be the accusative.
I can't give you a general rule (I don't know it!), but «на» with accusative works well only with «день»: «на сле́дующий де́нь» '(on the) next day'. It doesn't really work with other time expressions.
На + acc. can be used with other time expressions, but it has a different meaning: it works as a noun modifier. «Вы полу́чите отве́т на сле́дующую неде́лю» would mean you'll receive an answer that will be valid for the next week, "an answer for next week". This doesn't sound too natural with «отве́т», but sounds better with «календа́рь на э́тот го́д» 'a calendar for this year', «расписа́ние на сле́дующий ме́сяц» 'a schedule for the next month'.
I think this is vaguely consistent with the other uses of accusative vs. prepositional cases with "на": А location (on) is described by the prepositional case (на стене, на столе) while a direction (onto) corresponds to the accusative case (на стену, на стол).
Since the original sentence tells you when you'll get your answer (i.e., a "location" in time), the prepositional/locative case seems quite natural. BTW, a slightly old-fashioned "на дню" (today) follows the same pattern. The other cases you have mentioned (a schedule for the day, a calendar for a month etc.) indicate (vaguely) the "destination" of their applicability, hence the accusative.
How do you type the знак ударения and the Russian quote marks?
I use Linux with the default Ukrainian layout, formerly called Ukrainian Unicode (despite its name, it includes Russian and Belarusian letters too, although they're a bit harder to type than in the Russian layout). In Linux's Ukrainian layout, stress mark is AltGr+apostrophe, « is AltGr+б, » is AltGr+ю (and ы is AltGr+і, ё is AltGr+е, э is AltGr+є, ъ is AltGr+ї).
Default Windows layouts and Russian layouts on Linux don't include stress marks by default, so you'd need to install a different keyboard layout.
Russian keyboard layout in Linux is not as advanced as the Ukrainian one. I think Linux includes an additional option 'typo' («Включить дополнительные типографские символы»), which was usable with Russian keyboard layout, but I think it got deleted from the GUI for choosing it in the keyboard layout settings. You might have luck with setxkbmap -option lv3:ralt_switch,misc:typo, but I haven't tried it!
(I don't have Android/Apple devices so I don't know anything about these.)
I started the Ukrainian course but have been neglecting it. I’ll get back to it at some point. Now I’m focusing more on Russian, and using Windows exclusively.
"неде́ле"? I think, the easiest way to explain it is that the locative here conveys the location in time of this event.
In support of this hypothesis, similar sentences about hours can sometimes use the special locative form of "час": Вы получите ответ в пя́том часу́. (before 5 o'clock)
This one has «в», not «на». I think these pronouns should be treated separately.
But both "в" and "на" prepositions can be used with the accusative and the locative cases.
Could this also mean "the following week" as in "you will receive an answer the week after you send your letter"? Or does it have to be the week next week after today?
Does this mean you will get an answer "next week [not this week]," or "sometime during the next week?"
Translated this as "You will receive an answer the next week." Marked incorrect.
This reminds me of the middle school French teacher who always wanted me to answer as the textbook verbatim. What do you think?
To my ear, "the" is completely out of place in your sentence. Not being a native speaker, I cannot tell you with the 100% certainty that it's formally wrong, but I suspect so. The reason for my suspicion is that in such use, "next week" is an adverbial construction, not a noun. That's why it is used without any preposition, and for the same reason there shouldn't be an article. Compare it with "in the next week" -- now we are dealing with a noun. https://forum.wordreference.com/threads/in-the-next-week-vs-next-week.2509859/
Of course Duo wants you to give the textbook answer. Duo's a robot, it can't accept an answer it doesn't have in its database.