I hear that your daughter is strong-willed, impulsive, and challenging. She is also just barely 4 years old. This is when she begins to learn to manage her emotions. Until she can manage her emotions, she won't be able to manage her behavior. So everything you do in response to her "misbehavior" is shaping her brain, her emotional intelligence, and her ability to think about and manage her own behavior. So it's great that you try to be flexible and accepting of her emotions!
First and foremost, you should know that children develop self discipline when they give up something they want for something they want more. In other words, if your daughter wants to smash an egg, but she wants more to stay in positive connection to you, then she may be able to stop herself from smashing the egg. But this is not an ability that is already in existence in a three or four year old. It is developed through practice, over and over again, in making that choice between something she wants and something she wants more. That's how the pre-frontal cortex develops the neural pathways and neural memory for the child to exercise self-discipline in the future.
But here's the rub. If you FORCE your child to not smash the egg, she is not making this choice. Instead, she feels like a victim who was pushed around. And she gets gypped out of the opportunity to develop self discipline. That's one of the reasons why parents who punish actually raise kids who have less self-discipline.
But you can't just sit back and do nothing while your girl smashes all the eggs and breaks all her dad's tools. You do have to set limits. So what should your general approach be, to guide your daughter in such a way that she develops self-discipline and learns to manage her emotions and behavior?
1. Make empathy your "go-to" reaction to her, even -- or especially -- when you're setting limits.
Even if her behavior seems completely out of line to you, she has a reason for it. So, for instance, you see her throw the mop -- "out of experimenting or exuberance or something." You exclaim in dismay "Oh, no! Daddy's special mop!" You probably can't help yourself, and besides, this lets your daughter know that this is serious. But instead of turning to her in anger, you take a deep breath and remind yourself that she doesn't understand waste or money. She didn't know the mop would break. She was experimenting to learn, which is, after all, how we learn. What do you want her to learn here? If you punish or yell, she will go into "emergency" mode -- fight or flight -- and her brain's learning centers will shut down. All she will retain is that you are unpredictably scary, and when she experiments she should be sure to hide from you.
So instead, you go over to her and get down on her level and put your arm around her and say calmly "You were playing with Daddy's mop and you threw it. Was that fun? (You can smile wryly here. You are being empathic, not punitive.) Yes, I understand. You love to throw. And lots of things are good to throw -- balls, bean bags, even rocks if you throw them in the water. But Daddy's tools are not for throwing, because they CAN break...Let's go see what happened to the mop." Take her by the hand or pick her up -- lovingly -- and take her to the mop. Explore the mop, showing her the results of the throw. "See? Here's where it broke...Too bad...This was a good mop and now it is broken...I don't think we can fix this...too bad."
What happens next depends on your daughter. If she is crying, you hold her and empathize: "You didn't mean to break the mop. You wanted to see what happened when you threw it. I know." Let her cry as long and hard as possible.
If she is laughing, don't assume that she doesn't care. Laughter is a sign of anxiety and she is probably trying hard not to feel the guilt that would make her cry. You might say "You're laughing....But I know you didn't mean to break the mop....This is a hard thing... We need a mop, and this one is broken and we can't fix it."
More likely, she will simply be somber, looking to you to know what to do next. Now you want to introduce the concept of repair. "Sweetie, Dad will be sad that his mop is broken.....what can we do to make things better? You're right, we could buy him a new one, but that will cost money. So yes, we can plan to go to the store, so Dad won't have to make that trip. But maybe we should do some special chores for Dad, to help him, so he has time for other work? What do you think? Maybe for instance we could work together to sweep up in here, so Dad won't have to...What do you think? Could we sweep together?" Pick some chore that she can actually help with. Make it immediate. Don't present this as a punishment, but as a reparation. You are empowering her to make a repair.
Notice we haven't lectured her. She doesn't need a lecture to learn that breaking things has a cost. She's learning by seeing your concern, and by participating in the conversation and in the repair. Coupled with her desire to please you, all this will eventually will help her want NOT to break something even more than she wants to throw it.
2. Use play to help your daughter explore, laugh and connect with you.
Later, once everyone is calm, do some throwing with your daughter. You can get a bucket and some bean bags, and have a throwing session. Make this a really fun time with a lot of joking and laughing. You might consider every object in sight. Is that good for throwing? If it really wouldn't hurt something, let her throw it. So, for instance, she might learn that dish towels are okay to throw, but then need to be washed afterwards, and they don't go very far. Blenders really are not okay to throw, they would certainly break. Pans can get dented. Rocks might be okay to throw at the creek or in the field if no one is nearby, but are dangerous otherwise. Etc. The point is that you are helping her get this experimentation out of her system, so she doesn't have to experiment on her dad's tools.
Even more important, you are connecting with her by laughing, which helps you both release oxytocin, the bonding hormone. The more you laugh together, the more she will want to please you, so the more she will listen to you at other times.
Finally, roughhousing and laughing helps your daughter express and evaporate any anxiety that is driving her behavior. Your description of her often acting impulsively could indicate some anxiety, and laughter is the best way to address anxiety and help her get it out of her system. All young children have some tears and fears stored up, and anxiety is just another word for fear. So all kids need daily roughhousing and laughing just to clear out the anxieties of the day, so they can be more grounded, happy and cooperative.
3. Set limits when you need to, even when that means you have to pick her up -- and do it with empathy.
You say that now at bedtime , she runs away, hides etc. No child in their right mind wants to go to bed. So this is a predictable development. Have a sense of humor about it, but calmly, kindly, firmly, retrieve her and carry her to the bathroom. There is no reason to be mean about it. You can say, just as you do now, "It looks to me like you don't want to go to bed, Sweetie, AND it is time now." Then, give her her wish in fantasy, which the brain hears, pictures and reacts to as if it were real. So you might say "I bet when you grow up you'll NEVER go to bed! I bet you'll play all night every night, won't you?"
You say that once you "start brushing her teeth she settles in, and often then starts saying how she is tired. But is is very stressful for both of us." So I would suggest that there is no reason it has to be stressful. It is her job to test your limits so she knows which ones are firm. It is your job to keep a sense of humor while you hold your limits. Just accept that this is a stage -- she is testing to see if bedtime is firm -- and that you will almost certainly need to pick her up or hold her hand to get her into the bathroom. If she cries, that's okay, she'll be more cooperative once she gets it out of her system. If she laughs, that's even better. Either way, she learns that bedtime is firm and you mean it. Soon, she won't bother to run away. And she learns something even more important. She can't always get what she wants, but she gets something better -- a mom who understands why she doesn't want to go to bed, and who loves her through her resistance.
4. Support her to cry when she needs to by setting limits in a kind way.
Sometimes children will do something they know is against the rules just to pick a fight so they can cry. So while it is fine to be flexible and let her smash some eggs, it is also fine to decide that she has smashed enough. Often kids are looking for the limit. Ideally, when you decide to set the limit for her not to take any more, you need to be prepared to stop her physically. Then you can take her in your arms and kindly say "Okay, enough egg smashing, I won't let you do that" and hopefully she will burst into tears. Crying is what helps her get those tears and fears out that are driving her out of control behavior.
In this instance, you were too far away, so you were headed toward her but used your voice to say "NO! Don't take another" That's totally appropriate. But it is also true that she does not yet have a lot of impulse control. So when you say something, if she is already ready to grab the egg, it is really hard for her to stop. That will change as she gets older, of course. In this instance, if you can think quickly enough, you can call her name to get her to look up at you, and say "WAIT!" as you rush over. She still might not wait, but it is easier for her to pause, than to stop.
So what if she grabs the egg and smashes it anyway? You get down on her level and look her in the eye and say "I said NO! But you smashed the egg!" Hopefully, she cares enough about your dismay to begin crying. If she doesn't cry, you can say "I can't let you be in here because it is just too hard for you not to break the eggs." And take her out. At that point she will almost certainly begin crying. You hold her and resist the urge to lecture. You say "You smashed the egg, even though Mommy said No....It was hard for you to stop, wasn't it? We will try again tomorrow."
You did mention that she seems not to think what you say is important. If you set a lot of limits, it may be that you are setting too many and she just tunes you out. If you talk all the time, she may also tune you out. And, finally, if she is not feeling very connected to you, she may tune you out. Why would she feel disconnected? If she doesn't get to cry, she may have some big feelings stored up. That takes kids out of their hearts and disconnects them from us. So you might find that after she cries, she's more connected to you.
And that brings me to my last recommendation. Be sure that you are solidly connecting to your daughter. Listening to her. Spending some time every day engaged with her where you follow her lead, act as her assistant, and just pour your love into her. The only thing standing against your daughter's impulse to break everything in sight -- just to see what happens -- is pleasing you. Make sure that her relationship with you is the most important thing in her life, so that she gets practice choosing it above her other impulses. These days will pass, soon. Make sure you come out of them with a stronger relationship with your daughter, even if you leave a few broken eggs along the way.