"Normal people are asleep at that time."
Translation:Нормальные люди в это время спят.
I certainly don't know. But Latin also uses accusative for duration of time (admittedly perhaps not what is used in this precise sentence, but I think the matter you ask about more generally). And accusative pops up in German relating to time, too (https://en.wikipedia.org/wiki/Accusative_case). So there's some likelihood you'd need to pose your question to a proto-Indo-European speaker to get anything approaching an authentic answer!
It's beyond my Russian level to say for sure, but I don't think active participles are often used in that way. They're used to directly modify nouns, often rough equivalents of English subordinate clauses, not as predicates. Google finds nine results for "Нормальные люди спящие," and they all appear to fit this pattern.
Here's the first one, for example: "Нормальные люди, спящие днём, через 3-4 года становятся не очень нормальными." "Normal people who sleep during the day after three or four years become not so normal."
I'd recommend asking in the main forum if you have more questions about participles. There you'll find the people with the requisite Russian and English competence to give you authoritative answers.