Could you also say "Üniversitenin sınavları çok zor" and would this sound better or worse?
That would be "The university's exams are hard." It is ever-so-slightly different in meaning in both languages. :)
Is sınav instead of sınavlar also acceptable?
Looks natural to me, but I think the very best is to say "sınavlar".
I asked this because I've seen that singular is generally used in Turkish when talking about inanimate objects. But now I remember reading somewhere that if you talk about something in general (like here about exams), then you should use plural.
In some cases, they can look like natural. But yes, always try to use plural forms for situations like this.