"Whom" is properly correct in this case, but I'll concede that in colloquial English most people don't use it any more. I still do, though. Arguably, it's more important here than anywhere, as "kogo" literally means "whom", you actually can't use "kto" here like you could use "who" in English.
Yes. Also when 'kogo' is used in the question it is an indication that either Genitive or Accusative must be used in the answer. In this case ACC, GEN is used mostly in negations.
I guess in polish, sentence order is not as important as it is in English as the words themselves indicate 'what is doing what to what'? This is very much like modern German and even old English.
Can it mean "who is she seeing?" (meaning, who is she going out with like in a relationship)
There were loooong discussion under some comments about whom/who, some people say that 'whom' is too old-fashioned or even 'grammar-Nazish', some people claim that it's the only grammatically correct option... "who" should definitely be accepted and I believe it is.
"Whom" is the answer given, but it is virtually never used in simple questions, and you won't find it being used like this in EFL courses. Why not risk the wrath of the pedants, choose natural English and go with "who"?
But did "whom" ever work with the word "see"? Well in German it doesn't and English has strong roots in German languages.
It does work in German. "Wen seht ihr?" German doesn't mix up "wer" and "wen". Even though most speakers of English don't use the word "whom," I personally like it when dealing with cases in foreign languages. Kogo in Polish and wen in German refer to who is being seen (accusative case/DO), not who is doing the seeing (nominative case/subject)..
That's not English. it would have to be "who is she looking at?" but I think that in Polish there is another verb przyglądać się, which at this level we have not yet been introduced to.