I'm only a learner of Russian, but shouldn't it be Пишу о каждом дне? Shouldn't "o" take prepositional case?
Yes, you're right. That's correct, but it sounds a little bit archaic. (A lot of the Polish noun cases sound like the archaic use of Russian cases). I changed mine to the most natural construction--see above.
One of the suggested correct answers is incorrect, "I write of every day." This should read, "I write about every day."
Not necessarily. If the author is recording the events of each day, both 'writing about each day' and 'writing of each day' are correct, though the second is unusual and old-fashioned. If the author is describing how often he writes, then 'writing about each day' is correct and 'writing approximately each day' is, too.