The official proper accusative for ta is tę. It's an exception.
Some people say tą, but purists frown upon that.
I think it's because of the pronoun.. "tę kobietę" translates into "this woman". If you want to say "that woman", you need to use another pronoun like "tamta", so that would be "tamtą kobietę" .
What is not accepted? Te kobiety?
while te and tę are pronounced the same way, they mean different things. (but te should be usually accepted with pay attention to diacretics info)
but kobiety and kobietę are different words, and this is why your answer is marked wrong.
and here is one even better. http://mowicpopolsku.com/polish-alphabet-pronunciation/
yes, like what I've heard was "Czy on zna te kobiety?" - Does he know these women?
Audio is not perfect, but it pronounces kobietE even more audibly than it should.
Why is "He knows this woman?" considered the wrong answer? Surely "He knows...?" and "Does he know...?" mean the same thing right???
I don't know if you are a native English speaker (though I see you live in Manchester), but I guess using just your intonation/a question mark in an English question is just rarer, more colloquial and depends more on the context, while "Does he know...?" is the most common variant. In Polish they drop the "czy" more often, although it is still colloquial, but since it is used here, it is one more reason to use "does" in English. At least that must be the Duolingo staff's thinking.
Generally, when you form a question in English by only changing the intonation, leaving the syntax intact, you would express a surprise. If you want to do that in Polish, you can never include „czy”. Other cases of that kind of question in English are probably too colloquial for inclusion.
It seems like "czy" is redundant here and that some inflected form of "zna" could be used. Right?
Czy is not necessary, but often used. You could also say "On zna tę kobietę"? or even sometimes "Zna tę kobietę?"
if "Zna tę kobietę?" is the sentence, can I put "on" too?
Like this: Zna on tę kobietę? -> to specify the subject better as the verb zna can be followed by "he, she, or it" (if I am not mistaken).
Or is it not necessary?
"Zna on tę kobietę?" is technically correct, but "On zna tę kobietę?" is usually better. Polish doesn't form question by swapping the subject and the verb.