Hi C.,
I think you have a very nice and normal girl. She just repeats what she hears and tests your limits and perhaps you do not always read her cues correctly and not always consistent with your discipline.
What helps me when applying discipline is to always keep in mind WHY is my child doing what they are doing and WHAT do I want them to DO/LEARN and how can I teach them.
When your daughter is helping with the baby - praise her a lot. When she spontaneously does something good -praise and point out what you liked about the behavior. Ex: "I really liked how you were very careful about pushing that grocery cart without rolling it on peoples' feet!", "I like how you are very gentle with the little brother" , etc.
Your example with "We took her to the park today causd it was so warm and beautiful out. When we got home we watched some football and she kept kicking her father. " I think she was acting out what she saw - kicking. She just directed that at her father because there was no ball. This is the cue that needs to be read correctly and discussed, not just removing the child's foot and leaving her wondering inside her head "he doesn't want to play, but earlier kicking seemed to be fun". Questions like "Are you pretending that my leg is a ball?" or removing the foot and saying " Did you liked all the kicking we saw, do you like to kick some ball later with me?" would be more like addressing the underlying issue.
In re: to your time out and the fact that she almost broke the door. I've been through this technique only once. Never had to use it again. My little guy dislikes to be alone in the room period, and I think he was scared. Anyway, because of that I modified it slightly because I wanted my child to learn to behave and not to scare him to death. I warned him that if he didn't stop some obnoxious behavior (I think he was determined to paint the couch with red permanent marker at the time despite of my strong objections and offer of paper and more appropriate application of his artistic skills) he will be put in his room alone for time out. He chose to not to listen. I told him "Too bad you decided not to listen, I have to put you in the room now". He started protesting obviously, I added right away "You can come out as soon as you are ready to promise to be a good boy and listen to Mommy". That is the part where I TOLD him what I want him TO DO to get out of that room. He whined "But I do not want to be a good boy!", clearly we had no meeting of the minds here. I told him again while carrying him to the room "I will open the door and you can come out when you are ready to be good and to listen to Mommy". The room was in his sight now so he changed that song "I do not want to be alone there, I am scared!" Me: "Do you promise to listen to Mommy?". Him: "I don't want to!" I opened the room, put him on his bed and walked out and set right outside. In the matter of a second my child was at the door banging and screaming. I was talking to him through the door: "Darling, I really wish you can come out, I cannot let you destroy the couch and not to listen to Mommy. When you are ready to promise, you can come out right away." He screamed "I am ready!" I opened the door and got the evil eye child staring at me screaming "I hate you! I will paint the couch red!" . I said: "Oh, my. I can see that you are not ready. I am very sorry." I closed the door and set down again. He started the screaming again. I said " Where is my good baby? Where is my little boy that always helps Mommy and listens to Mommy and gives her hugs? Where did he go? I really want my beautiful boy back. " And so on and so forth. Many times I stopped talking because the child on the other side was louder that me. He said one more time he was ready, I opened the door again - and he was not. Finally, on the try #4, he calmed down, promised to be good and by his demeanor, I saw that he really meant it. I gave him a big hug "My beautiful baby is back? He is nice now? and where is the bad boy? You don't know? He left? Well, tell the bad boy not to come back anymore?" He was looking at me and said "I was bad, but now I will listen." I hugged him again and said "It is very nice to be with you when you listen. Do you want to do XYZ now?" Well, my point was that you don't just lock the child in and expect them to change. They are small, they do not know what you want them to do, they cannot read your mind, they do not know what is the social norm and they may not be aware at 3 y/o what they did to offend you. You need to use a lot of words and descriptions to tell your baby what is wrong, to tell her what you want her to say and to do to mend what she had done and to be loving through the whole process. Otherwise it becomes punishment and not discipline.
I never liked the soap in the mouth idea. From the child's perspective - you are hurting her. Grandpa does not have his mouth washed with soap, right? Because he is after stroke and may not realize what he is saying. Well, she is 3 y/o and I do not think she realizes what she is saying. She probably just likes how it rolls off her tongue like any funny new word and it gets a reaction from you on top of that. Dealing with these words, I would strongly tell the child "This is bad word, you cannot call Mommy/Daddy that." If she chooses to continue, time out is a good thing. Small one, like sitting on a chair or on a big bean bag in the corner. My son tried to escape those. I set with him, ignoring his pleas and counting aloud to sixty, holding him in my lap, if he interrupted me or kicked, or some other offense, I started from 1 over again. Boy, did my kid learned his numbers!
My son learned that the words TIME OUT meant business! You have to do time outs consistently at home for a while before your girl gets that she cannot get away with mistreating you. It may get worse before it will get better because she will test you and will buck against the rules you putting up. Be consistent and firm.
Now, I only mention a time out and my boy is yelling back "I am sorry, I will not do that again!" in a playground or on the play-date. Still, I always give him a chance to behave properly and to use a good judgment on his own. I always ask things like: "Are you running away from me?", "No I just want to look at this rock!", "OK, but what do you have to do if you what to go and look at the rock?" ,"Mommy, can I go and look at that rock?", "Do you want to go and look at it together", "Yes, Yes!". OR, if there is a cheeky monkey face and no response to my inquiry of what he is doing my next question may be "Do you want to go in a time out?" That gets his attention.
Sorry, it has been a long response. I hope you will find some of it helpful. You do have a great thing going that your daughter loves her brother. That is really great.Good luck with everything else.