"I see a cat here."
Translation:Я вижу здесь кошку.
I think it has to do with the new information. In Russian they tend to put the new information at the end of the sentence (although this can be overridden with verbal intonation). In this sentence the new information is that the thing you see is a CAT, so кошку goes at the end of the sentence. "Я вижу кошку здесь" would imply that the listener already knew you saw a cat, and you were telling them that the cat you saw was HERE, but that's not the intended meaning here.
Hopefully a native can chime in, but I suspect it's better, all else being equal, to keep здесь next to that which it most relevantly applies too. It's not important that "I" am here, it's important that a cat is seen here. So maybe the most natural place to put здесь is before вижу or before кошку. This is kind of how it is in English too, though maybe for a different reason, I'm not sure. It would sound unnatural to say "Here I see a cat".
The two correct answers put the location word in different places - "Я вижу здесь кошку" vs "Я тут вижу кошку". Is there a reason for this, or is it simply that those two sentences were presented? That is, would it be equally correct/idiomatic to say, "Я здесь вижу кошку" or "Я вижу тут кошку"?
Only inanimate male nouns have the same accusative and nominative. Animate male nouns, on the other hand, have an accusative which is identical to their genitive, hence кота. EDIT: Feminine nouns ending in the soft sign "ь" also have the same singular nominative and accusative. For inanimate nouns ending in ь, the plural also has the same nominative and accusative, but for animate nouns, regardless of gender or ending, the plural accusative is identical to the plural genitive.