"We are reading the newspaper."
I'm guessing Duo is teaching you that both 看 and 读 for newspapers are acceptable.
I was always told that 读 is involving reading allowed, or with your mouth, while 看 is reading with your eyes, or to yourself (notice the radicals on each character). I've also been told 读 is more equivalent to reading for academic reasons, while 看 is more casual reading. Anyways, I feel 看 is more natural in this case, but technically both work.
Both would make sense, but if you used zhengzai then you would be saying you are doing it right now.
Both are good. But I think the word 正 (“exactly, just”) does not give much emphasis here, unless there is something else to compare with.
Not sure if this is correct, but I think I remember my teacher saying that 正在 is used to really emphasize that an action is happening right now.