The apostrophe signifies here that something is left out, like in I'd, it's or she'll. What is left out is never written out in this case though, because it's outdated and out of use. 's nachts= des nachts. It's is the genitive, which makes more sense in expressions of time: 2 uur 's nachts = 2 o'clock "of the night" Actually, o'clock is "of the clock", so not so different, really. 2 o'clock o'night ;)
"Do animals sleep at night?" is a very general question and refers to all animals. "Do the animals sleep at night?" is more specific and refers to a certain group of animals. For example, I'm visiting the zoo and I ask the zookeeper "Do the animals sleep at night?" by which I mean the animals in the zoo.
Now I see why my answer (Slapen de dieren's nachts?) was marked as having a typo. I hadn't realised that the 's was separate from "dieren"! Whoops!
In English we can't just put the verb in the front to make it a question, like in German and Dutch. We often use the verb "do", which will make it "<Do> the animals sleep at night". I don't really know why, but i suppose just adding a do at the beginning makes it easier to make a question without fiddling with the sentence structure to begin with.
The spelling changes related to remnants of the case system are illustrated at https://en.wikipedia.org/wiki/Archaic_Dutch_declension#Case_usage