I read this and kind of chuckled. My daughter is now 4 and has always been very independent, strong-willed, and very much has her own ideas on how things should go and what she should do. I noticed it got particularly bad around 3..not listening, etc. My daughter is a Leo (August 17th) and I did some research on her sign and it fit her to a "t", so it made me realize that this is just going to be her tempermant to a certain degree. She is now 4 and it is not quite as bad, but if I tell her to do something like pick up something she dropped on the floor or her toys, it is a struggle. I try not to let it turn in to a power struggle as it very easily could (i mostly lose them). For example, if she throws a wrapper on the floor and doesnt pick it up, I have stopped picking it up for her and will leave it there and tell her to pick it up, if she doesnt then it stays there until she wants something and I tell her not until she picks up the wrapper (or whatever it may be). Then, I praise her and tell her "thank you so much for picking up after yourself and helping mommy..your such a big girl, etc.". It is kind of funny, becuase sometimes it feels like everything like that is such a struggle, that when she does pick things up the first time I tell her..I am like whoa..she listened.
Like almost everything else, I think this type of behavior to a certain extend is temporary, but I also beleive that how we parents deal with it now lays the ground work for how they react to us in the future with this type of thing. My main thing, is I try not to turn it in to a power struggle. It would be interesting to know if your daughter too is a Leo.
I am a 35 year old mother with a 4-year old daughter who is the love of my life, and I am sure will very soon give me many gray hairs :)