It's wrong for two reasons. Firstly, „tamten” is in nominative case, and this sentence expects accusative. Secondly, because „kobieta” is a feminine noun, you need to inflect „tamten” into feminine as well. Combining those two, the proper form is „tamtą”. Additionally, the last word is „kobietę” with an ę, but I assume you simply don't know how to type it yet, but it sounds like a normal e, so even some native speakers spell it incorrectly.
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.