It should be «кота́». This depends on the animateness: «Я ви́жу здесь сто́л» (стол is inanimate, it's not a living being, so its accusative form is the same as nominative form), but «Я ви́жу зде́сь кота́» (кот is animate, it describes a living being).
Animateness is something that doesn't depend on the context. It's a characteristic of a noun, not of a sentence. So, if you see a statue of a cat, you still use 'cat' as an animate noun even though that statue has never been alive. The same is true for dead animals.
There might be some exceptions (for example, the fish шпро́ты 'sprats' are usually seen canned, so it behaves like an inanimate noun), though. Also, some nouns can be either animate or inanimate (e.g. бакте́рии 'bacteria').
The distinction is not completely logical, because for example ро́бот is animate, even though most currently existing robots can hardly be described as 'living'. Ку́кла 'doll' is also animate (there's something creepy about ^^'), because when children play with toys, they address them as if they were alive.
For dishes, we have a different set of nouns that are inanimate: e.g. ку́рица 'hen, chicken' is animate, but куря́тина 'chicken meat' is inanimate. The same is true for ко́т 'cat' and коша́тина* 'cat meat'. Of course, there's nothing wrong with using ку́рица and ко́т to refer to the dish, but they're animate even when referring to the dish.
* No, this is not a dish in Russian cuisine. It's just that you can make up such a word from any animal name, e.g. роботя́тина would be understood to mean 'robot meat' and идейча́тина might be understood as 'idea meat', whatever this could mean.
We almost never say «могу видеть» in Russian. 'I can see' is almost always translated «я вижу», «могу видеть» is simply not idiomatic and sounds odd.
So, I believe 'I can see a cat here' is acceptable as a translation for «Я ви́жу здесь ко́шку» and should be accepted too. If it's not, it's worth reporting it.
@szeraja_zhaba Does this apply to other verbs as well? For example, would it sound odd to add «я могу» to any of the following:
To make this seem a bit more relevant to anyone reading this, the list you see above is a list of verbs that are among the 25 most used in the Russian language.
Also, is there a list of which verbs can/cannot pair with «мочь» in this way?
You can use "я могу" with any of this verbs, including "видеть". It's just that in Russian "могу видеть" would be interpreted with much more emphasis on "can" than on "see". "Я могу видеть (сказать/говорить/стать/etc.)" means that you are capable of that in general. It's a statement about your abilities, not about something that is happening right now.
This is a matter of English. In English, words that express sensing (see, hear, touch etc.) are usually paired with "can". "I can hear you" does not primarily mean that I have the ability to hear you, just that I do hear you. Translating to Russian you drop out the "can".
It will emphasise «здесь». You could use it for contrast:
— Я вижу там кошку.
— А я вижу кошку здесь.
However, when there's no contrast, such word order would sound odd.
Now I have to choose 1 from four options: Я вижу здесь ... 1 коты, 2 кот, 3 кошка, 4 кошку. Okay, 3 is wrong, because it's in nominative, and 4 is cleary right, becuase it is in accusative, but 1 and 2 can be BOTH nom. and acc., and should be accepted as correct. Or is this some flawed tricky question?!...
The accusative and the nominative looks the same for the inanimate masculine nouns and all inanimate plural nouns (and also for all neuter nouns and feminine nouns ending with "-ь"). A cat is an animal, i.e. "коты" and "кот" are animate nouns so the accusative would be different from the nominative. Namely it's "котов" and "кота" respectively.
I disagree with the answer to this sentence. Я вижу здесь кошку - is the answer to the question - what do you see here? I see here a cat. When - I see a cat here. Would be the answer to the question - where do you see a cat. Without a context, position of 'here' can be correct both before and after 'a cat'
Кот and коты are nominative (single and plural). Accusative for кот is кота. This one is accepted.
Inanimate nouns (nominative - accusative):
house: дом - дом (m)
dream: сон - сон (m)
code: код - код (m)
spoon: ложка - ложку (f)
book: книга - книгу (f)
door: дверь - дверь (f)
night: ночь - ночь (f)
Animate nouns (nominative - accusative):
dog: пёс - пса (m)
son: сын - сына (m)
cat: кошка - кошку (f)
man: мужчина - мужчину (m)
uncle: дядя - дядю (m)
aunt: тётя - тётю (f)
horse: лошадь - лошадь (f)
daughter: дочь - дочь (f)