I think you see things for what they truly are. I think your sister just tries to put labels on you.
Ok, you made a mistake by reading her e-mails. Don't do that again! You have to realize that everyone says things. The fact that you actually read them hurts. Of coarse it would. The fact that you feel the hurt and see how your sister is treating you all the time has nothing to do with your bipolar I think. You're just seeing her for who she truly is.
It doesn't matter what she told her friends...I'm sure they know how she is too! When you are with them, just be yourself and they will see that you are nothing like your sister describes you to be.
Also, don't look to her for advise or someone who you can confide in. Obviously she is just self centered. Keep up with your therapy appointments. That is what helps.
Make less visits with her.