My, your daughter sounds like a six-year-old!
Sometimes, our children go through very developmentally appropriate stages of exerting their independence. We are certainly familiar with the toddler "no!", which is one of the first manifestations of this. However, as children become older, their verbal skills flourish, and at this age your daughter is entering a season of reason and trying out her own logic. This certainly comes across as argumentative, but another perception of this is that your daughter is asking you to take her ideas (and she, herself) seriously.
If it were me, I would change the limits around this. When your daughter is challenging your request and you DO have time to listen, sit down with her for a minute and give her your undivided attention. Just listen to her idea without criticizing it, even writing it down on paper. Writing things down sends a signal to her that you are taking her seriously. Then, let her say what she needs to say. Unless her idea is unsafe or disrespectful/immoral, I wouldn't shoot it down right away. Instead, ask her "Let me think about that for a few minutes" and consider what she's saying. If we give our children "no" answers to everything and are always "right", they don't feel heard, just defeated. Then, decide yes or no, or suggest a compromise, and do explain yourself. Don't worry about "setting a precedent"--remember, she won't be in this stage forever.
Try to keep in mind that this really isn't a battle: it's just a question of what is important to you, and our job as adults is to guide our children by explaining why the "important" things in life are important. Being flexible about when we say "yes" doesn't actually send a message that Mom's a Pushover, but it does show our children that we respect and value their thoughtful work in trying to find their own solution to problems or requests. It's worthwhile to examine how often we can say yes to our children, and to do so whenever we reasonably can. Over the long-term, kids who get some "yes"s and some "no"s gradually begin to feel like it's safe to ask for parental approval because they won't be immediately shut down. (Very helpful in the teen years when they are likely to throw this back in our faces!)
Of course, there are certainly times that you need directions followed immediately, in which case you can say "This is a time for following directions. I need you to get in the car now. Once we are in the car, I'll listen to your idea and see if we can use it for next time."
One more thing I'm going to point out, and that's the last part of your request: I think we parents tend to believe that we should be unconditionally obeyed by our children. It's often what our parents expected from us. But it's a very high bar to set for our kids, both because of the fact that this questioning is an important stage in their development and because they are human beings too. If we put ourselves in their place, would we want to work for a boss who just gave directions without a reason why and expected them to be followed, even if they didn't make sense to us? Would we like to work in an environment where every suggestion was shot down? It's really not that much different. Yes, we hope our kids should respect us, but unlike the generations before, we are starting to see that respect is a concept that is best displayed mutually. If we treat our children in a manner that we would like them to treat us, they get actually get to "feel" what respect is; otherwise, it's more or less an abstract concept. I know that the Bible says to honor our parents, and we are also called to be judicious and thoughtful parents; I think that we can do both. We as parents can be the heads of our household and help our children be heard by honoring their developmental needs.
For more reading on the verbal exchange with our children, I highly recommend "How to Talk so Kid's Will Listen and How To Listen So Kids Will Talk" by Mazlish and Faber. An excellent book with very practical advice.