No, that's not normal, the clitoris and the stomach are in no way related to each other. Are you sure that the orgasm is what's causing the vomiting, it could be something else. Either way, drink some water and try to get your stomach flushed out. If you do believe that orgasming is what causes this then try to get an appointment with your doctor as soon as possible. That's only if you vomit every time that you orgasm. Other then that, try to eat food that your body can easily digest.
Absolutely not. Not unless she's reeeeeally ugly.
If you went "down town" and it really stunk...