Translation:I will write a letter to my friend.
As is mentioned below, the progressive tense (am writing) has a different form in Japanese, so it shouldn't be used here. This is more of a habitual or general present tense (I write every week, or whenever I go abroad, or that is how I keep in contact), or future tense as TanjaR8 pointed out.
I was marked wrong for:
I am writing a letter to friends.
I guess it's a little casual (I'd use this in conversation, much like "I'm writing letters to family"), but I couldn't really decide whose friends it was referring to (could be my friends, could be your friends, depending on context), so I didn't commit to a possessive.
Is this something I should report, or am I missing something?
If the subject of a sentence is left out in Japanese, especially in a situation like this, I think it is safe to assume the subject is 私[わたし]。
However, your translation is incorrect. "I am writing" (present progressive) would be "書いています" rather than "書きます" (present/future tense). The best translation is "I write a letter(s) to my friend(s)".
The particle に implies movement. Your friend isn't the one writing the letter in this sentence, but the one the letter is going to. "友だちは手紙を書きます" is "My friend writes a letter" (notice that the に is changed to the subject marker, は). Often, the subject is left out of Japanese if it is easily implied; Duolingo's sentence here has an implied "私は" at the beginning.
The present progressive "am writing" uses another form, 書いています. The one used here, 書きます, corresponds to a habitual or general present, or to the future tense. In real life, a tense change may sometimes be motivated by context, but we are here to learn the grammar so it would be a bad idea to allow it to cause confusion.
I am facing a big challenge with those audio tracks. I already found that I am not allowed to write whole sentence using kanji, which would be okay, however when I try writing it with hiragana only, quiz demands some of the names written in kanji. On the contrary, identical problem in different type of quiz accept various versions of the same answer. I assume that audio track questions lack those multiple variants of answers. Have anybody been thinking about doing some kind of mapping between 'answer-complete' questions and audio quizes that would extend answer pool of the latter? I think the source text of most of them match up, so it should not be a problem for an NLP enginneer ;)