I do not know anyone who would actually say this. About the closest would be something like "we know who the owner of the newspaper is," or maybe "we know what the political position of the newspaper is."
I realize we haven't gotten far enough yet to build such complex meanings, but maybe "Wir kennen das kind" might make more sense for the skills covered so far? (Just a suggestion.)
why is this course obsessed with translating nonsense sentences like this one? and, another example from many, 'the bear reads the books' surely they can think of actual normal sentences for us to translate. Do they do this to throw you out, like trick questions, or are these actually things german people would say?
If you ever have a problem with an answer being unexpectedly rejected, it helps the most to post your full sentence, exactly as it was written. Otherwise we have to guess what else you wrote. "We know the newspaper" is, as you can see here, the suggested answer.
My tip is to make a screenshot of any unexpected errors you receive. You can link them in the discussion page as 'evidence', or later submit them as bug reports. It also helps when you have an unjust rejection, that you click the report button "my answer should have been accepted". These are the best ways to fix such problems :)
German distinguishes between wissen (to know a fact) and kennen (to know someone, to be acquainted with someone or something).
As do many other European languages (e.g. French savoir, connaître, Spanish saber, conocer, Esperanto scii, koni, ...).
Here, die Zeitung is not a fact that you can wissen; it's a thing that you can be aware of (or not), that you can be acquainted with (or not) -- so kennen is the appropriate verb in German.
Yes, I think you are correct. You could rephrase it as "We are familiar with that newspaper." It would not be clear from the unconnected sentence, however, if the speaker is aware of the newspaper only by name and/or reputation or if the speaker knows the newspaper because they read it.
I tried: We know about the newspaper. Duo has accepted "know of" as a correct translation of kennen in other exercises, but it seems more natural to me to say "know about" rather than "know of". However, it may be that another verb construction is necessary to say the former.
My problem with the listening exercises is that if I can't understand something, like the subject or the verb, I am supposed to use context, but which context? How do I know which part I'm mishearing and which part is correct for me to make the decision around? (Sorry for the poor English, I'm frustrated.)
Why is it not accepting "We are knowing the newspaper"
Because that is not correct English.
We don't use the -ing form with verbs such as "know".
At least not in the form of English that Duo uses (based mostly on US English).
You might hear "I am not knowing the answer" in, say, India, but that's not considered correct in the US and would not be accepted on Duolingo.