H.W.
Does she have her own room? I think for rudeness, that would be one simple way to address it: "I hear your voice is very rude/mean. You may go to your room until you are ready to check in with your brother." This puts the onus on her to either check in with him ("Are you okay? I'm sorry I said that." etc.) immediately or take her attitude elsewhere.
When she demands: "I don't do favors for little girls with snotty voices. Would you like to ask that again in a pleasant voice?"
or "Please go to your room. You may come out when you've found your pleasant voice."
One thing to keep in mind is to stay as emotionally neutral as possible. If she doesn't leave to her room when she's being rude, simply take her hand and walk her there. "You may be as rude as you want when you are in your room with your door closed. Out here, it's expected that you may speak to people in a polite way. Do not come out until you are ready to do that."
I do think it's a stage, and they're sometimes 'trying on' the attitudes of the older kids they see at school. I do also think that "I'm not listening until you use a nice voice" still gives *her* the benefit of getting your attention while being rude, which is why I would choose to have her leave the common areas and be alone until she can regulate herself in how she is asking for things. I would also say that at 6, if there's any name-calling of other people, she needs to either check in/apologize immediately or leave until YOU are ready for her to come back. "Go in your room until I call you back. Your rudeness is not welcome out here right now."
You have to model as much positive behavior/talk as you can. Be sure you are asking please/thank you, checking in, apologizing when necessary, etc. Make sure to check your own language when you are frustrated. Our house rule is that 'stupid' isn't our favorite word, but calling one's own inanimate objects 'stupid' doesn't get chastised; calling another person or their actions 'stupid' does.
And for what it's worth, I do volunteer in the library at my son's school on the day the first graders come in. Wow! Some of them do really have mouths on them. I did interrupt two girls who were insulting each others hair ("Bows are stupid." "Nuh-uh. Buns are stupid.") and insisted that "You may talk to each other kindly, or walk away from each other." Just to say, they do hear it at school, certainly!