"You have the right not to answer the questions."
Translation:У вас есть право не отвечать на вопросы.
Thank you so much for your help. I have reported dozens of errors since I started this course 10 months ago and I have NEVER received not even one reply from Duolingo. For me, the author of this course finished his/her work and disappeared. It's thanks to people like you that I learn and I am very grateful for that.
Sure, I would mind ;-)
Jokes aside, Russian verb "отвечать"="to answer" is (usually) intransitive, unlike its English counterpart. (It has some uncommon and fairly non-standard transitive uses but you should ignore them - the chances you'll ever come across those even after mastering the language are basically zero.) As an intransitive verb, it does not take direct objects without propositions. The two most common/useful prepositions are "oтвечать на вопрос/звонок etc." = "to answer a question/phone call etc." and "oтвечать за [что-то]"="to answer for something" or "to be responsible for something".
Now to the second question: it's accusative. Having established the necessity for "на", recall that "на" can take either accusative or prepositional in Russian. In it's typical uses, "на" takes accusative to indicate direction (onto) and prepositional to indicate location (on, on top of). While it takes accusative in this example is less clear since "на" is used "figuratively" here, but accusative is the typical case for such figurative use. If you were to try prepositional instead, it would sound like you are answering while sitting/standing/lying on a list of written questions.