Does Duke consider the ACT writing section? I know that it is not required but I feel as though a 35 composite with a 7 essay score could hurt my chances somehow.
The writing section is recommended but not required for admissions. Thus, I would not worry about the writing section with your 35 composite score. More details are listed here: https://admissions.duke.edu/application/instructions#testing
In the last few years nearly all colleges have started to phase out the standardized testing essay component. Don’t worry about your score—nobody even knows what a good score is, so admissions won’t be giving it much weight.
It won’t hurt your chances for Duke. Admission officers understand that the essay was something you wrote quickly under a time pressure, and it will not necessarily be reflective of your actual writing skills. They will weigh your application essays more heavily. I know people who got an 8 on the essay and still got into Duke.
I, personally, believe that your CommonApp essay and writing supplements are incredibly more important than any score (especially a writing score on the ACT). However, if you feel anxious about it and Duke is near or at the top of your list, you could take it again (if the cost isn’t an issue and you don’t mind spending a Saturday) to improve your writing score AND possibly bump up your superscore to a 36 as Duke does pure superscore. I took the ACT after applying, knowing that Duke would be the only school to take it and against all advisement from friends and family because I didn’t want to regret anything and my superscore went up by 1 point. Did it matter? Will never know.
Bottom line, I think it won’t matter in the long-run but I understand how anxious it could make you feel (I would be the type of person to let something like this kill me) and how it could be worth it to give it another try for peace of mind.