Sort of. But not quite, from what I know from DutchGrammar's website, the des ('s) translates into the old form of the :)
But keep in mind:
's morgens = in the morning
's middags = in the afternoon
's avonds = in the evening
's nachts = in the night (for a short period of the night) / at night (for the whole period of night time)
Yes. That is an incorrect translation. We are being that the student works at night, but we are not being told how many nights the student works. So, for instance, we do not know if the student works 1, 2, 3, 4, 5, 6, or 7 nights out of every week. If we were meant to get the idea that the student works every night, we would see the word "iedere in the sentence". I'm not sure, but I think that if your translation was the correct answer the Dutch sentence would have to read something like the following: "De student werkt 's iedere nachts."