I thought it might be kočka, since kočku would be accusative, would it not?
It is "kočku" (in the accusative, as you said), because it is the direct object in this sentence.
Can't it be both nevidíš And nevidíte?
Yes, both are accepted. Nevidíte is singular formal and plural ;)
I wrote 'nevidiš kocour' and was marked incorrect. Not sure how many 'cat' alternatives to accept, but I was mildly annoyed.
"Kocour" is fine, it's a male cat, but you need it in the accusative for this sentence to be correct. -> "Nevidíš kocoura."
Here in Duolingo is both of them correct: Nevidíš kočku and Nevidíš kočka. I have thought, only kočku is correct = Akkusativ. ??
There is no acceptable translation for this sentence that uses kočka. Perhaps it was a Duo hiccup if it showed you one.