"Plants don't have intestines!"
Translation:Pflanzen haben keine Därme!
Yes, you're right. This one is kind of strange and you wouldn't say it like Duo suggests it.
No. "have" is not an auxiliary, but a full verb here. And for negating a full verb you have to use "to do".
You could say "Pflanzen haben keinen Darm," just as you can say in English "Plants don't have an intestine." The switch to the singular may be considered colloquial in German and English, but it's common, and I suspect Duo accepts it. However, since the English stays in the plural, I think "Pflanzen haben keine Därme" is a better translation.