Translation:You have sixteen seconds for the answer.
odpowiedź is noun so respond may be not ok. Hint may be for odpowiedz - 2nd person singular, imperative, instead odpowiedź - noun - both mean answer. for example now I have English DUO - I hit reply/ on Polish version I hit odpowiedz.
I think response/reply/answer are probably all ok. On the other hand I don't know if one can give response or reply in 16 seconds.
You're right that "mieć" indeed takes the accusative, which is why the "szesnaście" is in accusative case. However, if a number ends in a digit other than 2, 3, or 4 (with the exception of 12, 13, and 14) you have to use the genitive plural of the noun the number refers to.
So it would be masz trzy sekundy (you have 3 seconds), masz siedem sekund (you have 7 seconds), masz trzynaście sekund (you have 13 seconds), masz dwadzieścia dwa sekundy (you have 22 seconds) etc.