Does she actually want to do the DofE? My dd was given the opportunity via school to do it, but it was too onerous and she decided she didn't want to do it. I'd let your dd drop that. She's obviously not enjoying the challenge.
As for homework, well. It needs to be done and she knows that. What she is fighting against is not the homework itself, but you telling her to do it. So stop telling her. Maybe mention on a Friday evening that she might need to think about doing her homework at some point over the weekend, and leave it at that. She's the one who's going to get it in the neck from school if she doesn't do it, so casually allow her to discover the consequences for herself.
If she isn't bothered by being grounded or not having screentime etc, then you are wasting your time using it as a punishment, aren't you? You say she is stubborn. The way to cure that is for you to avoid giving her anything to be stubborn about. She is pushing at boundaries so move the goalposts and give her more autonomy, so she hasn't got boundaries to push against. Give only praise when she is being pleasant or co-operative, and ignore everything else as best you can.
Maybe sit down with her tomorrow, and say that you've decided you are no longer going to tell her to do her homework, and that she's growing up and is ready for more responsibility, so you are going to leave her to it. You could also perhaps ask her if she would like to stop doing the DofE. Her choice.
Tell her you think you've been being too stern with her, and grounding & other punishments are cancelled forthwith. Tell her that you don't want to fight any more. In return, you would like her to co-operate with the new regime and accept the new responsibility; and that includes doing the occasional household chore when asked. Say that you are sure that she too wants to live in a happy home.
Chuck her in at the deep end and see how it goes.