How do I let a friend know that her behavior is unacceptable without hurting her feelings?

Simply the only way to go about it is frankly and with respect. Yes she might be hurt by the reality but as long as you are respectful she might change for the better. Good luck!

To be honest, I think you just need to be straight with her, tell her that she is being rude and being racist and that she is the only one out of the whole group that acts that way and you and your friends are uncomfortable with it

I would suggest focusing on one part of the issue at a time. Try dealing with the racism first. Understand exactly what you're going to say to her and how you're going to explain it. Don't focus on any of the other issues until you have this one down and she comprehends her stereotyping and her privilege. Explain from a logical standpoint as best you can, keeping your emotions out of it as much as possible, and be patient about it. People who refuse that sort of criticism often don't understand and/or cannot fully understand what it means to not be a member of a dominant social group and feel hurt because they have never considered themselves to contribute to a problem.

Go from the worst issue to the easiest issue to deal with. As odd as this may seem, it will feel less hurtful to her if you focus on issues that aren't minor. It will not feel like nitpicking at her personality and attacking her identity if they are closer to legitimate issues instead of small personality quirks.

You may have to be patient with her. No matter how gently you try to break these things to her, there is still the possibility that she will lash out. It's expected--no one likes it when they feel like they're being attacked. If she needs alone time, give her it, but remind her that you guys still care for her. Be completely honest--lying or watering things down will only make it worse--but be prepared to do a lot of reassuring and comforting. It's hard to accept criticism and heal relationships if you don't think you've done anything wrong.

I agree, but I think it might soften the blow a little if you open with something like: "We all care about you and think you're awesome. We need to talk about some things that bother us so that we can all continue to be friends." If you have too many people; however, she'll feel ganged up on.

Your friend is boasting some heavy narcissistic tendencies. Be certain to tell her that her opinions and personality are fine, but there is a difference between being engaging and being hostile, and she steps across that line. Tell her that that is a line you do not enjoy being crossed and if she wants to continue being your friend that she has to respect your boundaries.

Remind her that you enjoy her friendship, but just as she is free to voice her opinions, others are free to find her opinions obnoxious and rude and respond to her accordingly. Our freedom does not excuse us from the consequences of our actions; she is free to be loud and racist, but others are free to ignore her and judge her based on her actions. We are each responsible and accountable for out own behavior, and if she does not take accountability for her actions, then she is just being a social bully and a jerk.

If she will not listen, then stop concerning yourself with her emotions. If she can not extend the courtesy of being considerate then she does not deserve the courtesy of being considered.

Thanks for all the different perspectives. This has given me lots of ideas about how to better deal with this issue, thanks :)