Would this also be the way you would say "The birds sang about the spring"?
Is there a simple way to say either one more clearly, so there is no confusion?
Perhaps sang i løpet av våren for sang in/during spring. I don't think there's a clearer way to say "about spring."
Would one translate "in spring" (without the article) as "om vår"?
Nope, still "om våren".
Why is 'sung' marked as wrong?
Because it's the past participle form and what you need here is past simple (sang). HTH