Your child isn't autistic. She is pushing your buttons to get a reaction. It's ugly and irritating, but it's working. She's getting under your skin by showing her butt here, and what you have to do is change how you react if you want to change her behavior.
It sounds like you are home with her all day. As hard as that is, it's the way to put her through behavioral bootcamp this summer to get her to stop doing this stuff.
Put something on the refrigerator that shows "strikes". You give her 3 strikes and then she is out. That means when she starts talking back to you, you say "That's strike one." She continues and you say "That's strike two." She'll escalate and you say "That's strike three and you're out." Take her to her room and shut the door. Let her have her tantrum inside her room. Don't let her come out until she calms down and starts to behave. The first week she will probably in her room more than she is out of it, and that's okay. She will probably scream a lot, and you'll just have to accept that. Don't let the crying and screaming change your stance - you must ignore it. Make sure that she is not alone with the baby. You must protect the baby from her while she is angry and jealous. She is too young to trust to not strike out at the baby in anger.
When she is talking nice to you, doing what you ask, compliment her. Tell her what a big girl she is and how proud of her you are. Give her some special treats that you have at the ready. Smile and sing and talk to her. As soon as she turns the tables on you, go back to "That's strike one" mode. Keep your voice and your face the same each time. And under no circumstances do you allow her to get out of having to go to her room.
You must be 100% consistent. You must show her that you mean business. Don't bother to try to explain a lot of stuff to her. It isn't doing any good telling her that she makes no sense to people. She likes it because it gets a rise out of you. What you have to do is make her lose the privilege of being with you or you paying attention to her by putting her in her room.
If she messes her room up, take everything that she has messed up and remove it from her room. Say nothing. Just continue to do it. At the point she is missing all her favorite things and her room is bare and unhappy, she will start to understand. When the tide starts turning and you actually SEE that she is thinking before she talks or acts ugly, when you see that she is trying to control her behavior, then you know that you can start offering back her toys/things one at a time because she has earned them by behaving well. You can do it up to like 3 times a day. "I liked how you handed me a diaper for the baby this morning and you have talked nicely. What a big girl! Here's your Little Pony." But on any day that you have to give her 3 strikes and send her to her room, you give nothing back to her that day.
This will be really hard while you wait for her better behavior to kick in. But you must persevere. You just must. If you don't, she will be a holy terror in school, and this is your responsibility to fix now.
When things are getting better, make sure that you carve out time to be with her. Quiet moments when you read to her, do a quick art project together, etc.
Wait for at least 2 weeks to take her out. You need to get ahold of her behavior at home before dealing with her at the park, library, grocery store, etc. If I were you, I would wait until your husband gets home to go to the grocery store and run errands.
When you do start taking her out, strap her in the carseat and sit down in the back of the car before leaving the house, and look her straight in the eye and say "If you behave badly while we are out, I will take you right home and put you in your room. Do you understand?" Take her somewhere as a test that is easy to corral her and get her back in the car. She will likely test you just to see if you mean what you say. You HAVE to do it, mom. Be hard-hearted so that she knows you mean business. If you have to leave a cart full of groceries, you just do it. If she's been at the park for 5 minutes, you just do it. If you have ordered food at a restaurant, tell them you need it to go and go stand at the front door with her while you wait for the food and then put her in the car. No eating it at the restaurant.
If she starts screaming and crying when you strap her in the carseat, stand outside the car and act like you don't care. "Read" your phone. Every few minutes, turn to her and say "Are you done yet?" Eventually she will calm down and then you drive her home. This teaches her that number one, you will not allow the bad behavior, number two, if she pitches a fit in the carseat, you don't have to put up with it and sit in the car with her (it's not really safe for you to drive with a screaming child in the car), and number three, she WILL end up in her room when she gets there. There are consequences for putting her family through her drama, and you must give her those consequences in spades.
Don't take her anywhere that it's too important to be on time or that you have to do something so that you can't follow through on your behavioral modification plan. It's so hard, but you just must do it. Eventually she will get this through her head. It will take all summer and more. Have your husband on the same page as you, or she will play you two off each other and try to manipulate him to her will.
She will eventually learn the lesson. It takes lots of patience on your part, and will only work if you show 100% consistency. Children need limits. They need boundaries. If you don't give them to her, she FEELS out of control and will act accordingly. It seems cruel to do it over and over, but it is training that your child needs and actually wants, no matter how much she fights you.
You can do this. Make it your mission.
