I went under for wisdom tooth extraction, so maybe I can help! In my experience, it didn't make me nauseous; I've heard it can for some people (about 30%), though, so I can't tell you for sure whether it will or not.
Going under seems terrifying to think about, I know, but maybe you'll be less scared if I share my experience? :D
So first, they put some heart rate and blood pressure monitors on me (no needles)!
Next they gave me some laughing gas through a nose mask, which made my feet feel super weird, like they were somehow both the weight of concrete and floating like feathers, but it didn't hurt or anything.
Then they used a small needle, very quickly, to put in an IV. The wound was so small that I couldn't see it at all the next day, and the pain was half that of a vaccination.
And if you're worried about the prospect of saying weird things under anesthesia (I sure was!), don't worry, it's not like being drunk or losing your inhibitions or anything. So I was lying there, then they must have put the IV anesthesia in, because poof! out like a light. It wasn't even like fading out slowly or anything. I was tired and loopy, but awake and in my right mind. I don't remember falling asleep. Next thing I know, I'm waking up and they tell me it's all done! Though, to me it felt like it had been about two minutes since I dozed off.
In the car home, I was tired and out of it and having a bit of difficulty with keeping my balance, but I wasn't nauseous, and I was also fully... what's the word? Sober, maybe?
And finally, by the time all of the induced numbness in my face wore off, most of the pain was gone anyway. It kind of felt like that pain when you've been spending a lot of time using your molars for crunchy foods, like if you bite popcorn kernels or eat a bag of nuts. But it's not as bad as you might hear! Honestly, it's nowhere near as bad as period cramps. XD
So, in short, nausea is unlikely, the pain is definitely bearable, and you don't need to be worried about spilling all your secrets to the surgeons. I hope this clears some things up!
Good luck with your procedure, and wishing you a speedy recovery! :)