I'm a little confused. It's masculine, so the singular genitive isn't the same as the plural accusative, as it is with feminine nouns, except mężczyzna, which becomes mężczyzny, like a woman, because it has a feminine ending. Kota has the -a feminine ending, but doesn't become koty in the genitive case?
Nominative is kot, it becomes kota in genitive and accusative.
i do not know why you think nominative is "kota"
I think you might have missed the rule that masculine nouns divide into animate and not animate, where animate have accusative=genitive and not animate accusative=nominative.
Thaaat, might be it. I got that animate and inanimate masculine nouns exist, but I think I've gotten the second part switched around, and then ended up losing sight of kota as the genitive and accusative, but not the nominative, which is where my confusion arose. Thank you, sir.
I think I have missed something. I've never seen any rules in duolingo. Where are they?
Depends. There are tips and notes in the first lessons (until adjectives 1), some people might be unable to see them because of "experiment".
After that there are no tips and notes. I might have used wrong expression, but I think there is talk about "animation" in the comments for lessons before this. (but only comments like this one, not real notes with rules. )
Widzę or widzem... Duo has said widzem is correct... is it actually wrong, but I am getting a free correct answer?
By the way, what is the differnce between, for example, "Ja widzę" and "Widzę"? How often and in what situations do you use pronouns in such sentences in Polish?
They're redundant. Unless you want to stress the "I" part, there's no need to say "ja", or it may even sound weird, although technically correct of course.