Picking your battles and not getting your point across and/or as a parent feeling like you're not getting respect after letting a handful of items slide is frustrating, but it can also breed an increasing frustration and resentment.
My two cents: It may be helpful for you (and DD) need to evaluate your dynamic. Begin evaluating what's inciting the conflicts, look at how you react and also how she responds to your instructions/authority. Where is the issue? Is it truly a power struggle? Does she feel you dictate chores/tasks but don't engage with her in many other positives aspects?
An issue was a scooter versus a bike... pointing out to her that if she wanted to ride her bike she should stand up for herself. In this case, this is what you wanted was to guide her and encourage her to do what she actually wanted... You saw it as a teachable moment. However, it seems she felt you pushed her to stand up for herself, when in reality she just wanted to do what her friend was doing. There's give an take in relationships. It is possible that she was navigating her friendship, but in a different way then you saw the situation... I'll ride my scooter this time, but maybe the next time she played, she would have ridden her bike, or it might have taken her 5 times before she felt comfortable with riding the bike.
Each child will get to each milestone in their own time and in their way. As parents, we want to seem them succeed, be strong and confident. As children, we want to fit in with peers. These instances with her might need more encouragement in a manner that she accepts as advice versus the more parental role of "do this/do it this way".
If you can step back and review the situations which arise, you might find you learn more about her... what gets her to respond as expected/appropriately and what pushes toward disrespectful. These indicators can help you tweak your interactions toward reinforcing your goals of raising and guiding a confident young girl through the awkward hormone filled years, while also achieving the respect you deserve. The bonus will be a stronger, happier relationship for both of you.
It's so important to talk to kids (and anyone we have deep relationships with). By investing the time to openly discuss expectations, behaviors and consequences, we open the door to starting anew -- sometimes we just need to find a slightly different course or a new way and have a clean slate.
Things will always still come up, but a simple phrase to remember is "correct with love". Privileges can still be taken away and earned back... if you've discussed and agreed slam a door = no door, talking back = no phone, etc., then she knows the consequences and it's her turn to fulfill her side of the relationship.