Peg has described the difficult 2 yo so well. I won't add to that but I'd like to add something I learned when I was learning to be a supervisor. Focus on the goal instead of on finding a way to show that I'm in charge. I was, for the most part, supervising men many of whom did not accept that I, a woman, could be in charge.
Your goal is probably to have a more do-operative little girl. How can you gain that. It isn't by being the boss. Once we, a parent or a supervisor, fall into that trap we've started a power struggle. As you've seeing, the kid (or the employee) wins because we get so frustrated and start to "give orders" instead of focusing on what "they" need to get the job done.
I'd back off completely from nap time. She may not need a nap. You need her to have a nap so you can rest. Let's find a nother way. Changing the pattern that you and your daughter have created will take time.
I think the first thing to do is choose a couple of "rules" and decide on consequences for those. As I've heard alot of people say, "choose your battles." Misbehaving in public by pulling your hair would be a situation for which I would immediately leave the library with her. Tell her in a calm, assured voice, that even tho you know she'd like to pick out a book we have to leave because you are hurting me.
It will take several times of this sort of exchange for you to see progress. In fact her misbehavior may intensify before it gets better. That's where consistency comes into play. You're right, "she's testing." She's not so much testing you as testing her world which is expanding. What are the limits?
Then find ways to spend pleasant time with just her. If whatever you're doing isn't pleasant, tell her you'd like to do such and such with her or even, you'd like to share some fun time with her but this isn't fun and leave. Stay within her sight but don't pay attention to her. She is insecure with the new baby, and her new abilities, even tho it doesn't seem like there's been much of a change for her.
Baby is held. She needs to be held too. It's not that she just wants to be held. She needs to be held. Try to see what is happening from a 2 yo's view point. I think it helps to actually get down on my knees and look them straight in the eye. This is difficult to do when you are holding a baby or carrying one in a sling. You can move to a chair, sit down, and ask your daughter to come to you.
I find myself getting impatient with my grandchildren because I have goals unrelated to them. I want to get the dishes done before I start dinner. Then there's the clothes to fold. My daughter gets upset when she can't get the vacuuming done. So we push the kids away without really listenting to them. I say, just a minute, until I get these last few dishes in the dishwasher. After a few times of this they act out. They will get my attention one way or another. So I've learned to stop what I'm doing, squat down, or lean over now that they're older, and really listen. Then, I say, we can do that in 10 minutes. They watch the clock. YOu can set a timer for your 2 yo.
There are all sorts of skills to learn so that our babies and children will be more co-operative. It helps my daughter to remind herself that it's the kids job to test. They have to find out where the boundaries are. (I tell her it's my job as grandma to spoil them. She doesn't quite get that one. :):))
Peg gave you some sources of amazing ways to get along with 2 yo's. I'll add small book that gives ways to deal with various situations in very few words. I found this book at the library and liked it so much I bought it. The title is off putting to some parents. How to Con YOur Kid, Simple Seams for Mealtime, Bed Time, Bathtime----Any Time, by David Borgenicht and James Grace. This book already assumes that the parent recognizes the child's developmental level is that of a child and that they are respected. Because children are not adults they are motivated in different ways than adults are. The giving a choice between 2 things is an example. Do you want to put away your toys first or make your bed first? When the toddler is 2 you know that you will have to help You've given them a sense of control over what they do while limiting it to what you want them to do.
I admire all of you who have a toddler and a baby. That is a very difficult time for everyone. As a grandma I'm fortunate in that I can sit down with them and not be concerned about the dishes, dinner, or the laundry. And best of all I get to go home. :):):)
One of my favorite sayings is "this too shall pass." when I had more responsibility I'd add "if I live thru it." I always have.
I reread your post. re: naptime-try going without a nap. Give her some toys, books, put on some music and tell her she doesn't have to take a nap. She can just have a quiet time instead.
re: bedtime. My daughter and many others don't agree with me. I always lay down with my grandkids. My granddaughter was nearly 3 when her baby brother was born. My laying down with her, gave me a rest, gave us both quiet, loving time together. And she went to sleep much faster. Then of course, I don't have the dinner dishes waiting for me.
My daughter just kept putting her back to bed; the same with "baby" brother. She wasn't consistent. They still learned in a few weeks. What is needed is consistency in always doing it and consistency in being calm and matter of fact. That's difficult. I'd rather just lay down with them.
Music or a story on CD or tape helps them get to sleep. Of course a nite lite and their bedroom door cracked a bit. The bathroom light left on, a cuddly toy and a blankie. And then a firm walk back to bed holding their hand. Never give up. Do it all night with firmness and lack of anger. That's the really difficult part. When we get upset they get upset. It's easy for us to get upset when they're upset. However, we are the adults and have the capacity to learn how to remain mostly calm.
If you want to get her to sleep in her own bed and stay there one of you take care of the baby so that the other will have no interruptions. When I saw it done on Nanny source the mother stayed in the bedroom with her toddler and just kept putting her back to bed the minute she got out. And she did this without saying a word. Nanny said you might have to do this several nites in a row. This mother did it 2 nights if I remember correctly.