Perhaps you could say "I am the father" by rearranging word order to emphasize the word "pater"? I would imagine that would be something like "Pater ego sum", correct me if that's wrong.
And then maybe "I am a father" would simply be "Pater sum", without the pronoun, to make it sound more general?
It would indeed always be more clear in a context what it's exactly supposed to mean.
It's also correct (and accepted, though it may not have trickled down to the user-side grading yet, it can take a while). What's most natural depends on the context and the desired emphasis. The pronoun is generally used for emphasis ("I am the father"). The verb is most commonly placed at the end, but if you also want to emphasise "father" it's perfectly fine to put it there.
The button in the lesson provides no opportunity to say what is wrong with the audio. How can we do that, other than in the sentence discussion? How can Duolingo be sure that a re-recording of the sentence corrects any problems if they have no description of what the perceived problem is? How else can we know whether a problem with the audio has already been reported, and what the reporter considered the problem to be other than through the sentence discussion?