It's always scary when your kids need to have any procedure done especially when they are so young and you can't really explain to them so that they will understand.
I'm a mother of 4, one is a special needs child and I'm also a nurse. Believe me, I'm the most paranoid mom around. I always think the worst.
My kids have had many procedures done at Children's and also several long hospitalizations. I would be confidient in the care that they give.
Two of my kids have had their tonsils and adnoids out. My youngest had his out at about 14 months of age. He had sleep apnea so bad that I also couldn't sleep afraid that he would stop breathing and die. It was torment to listen to him stop breathing and wait until he started again. He did end up having to spend the night in the hospital as when kids have sleep apnea so bad, their chemical receptors change and they get used to having a low oxygen content in their blood and the trigger to breathe is triggered not by low oxygen but by high carbon dioxide. So when these kids have a good air exchange and good oxygen content, they forget to breathe. It takes a little time for their bodies to reset to normal. I hope that makes sense. It is not common but can happen.
Also, it is so much better to get this done when they are young. From what I hear and have seen, the older you are when you get these out the more painful. My oldest was 4 when he had his out and he almost had to be readmitted for hydration as his throat was so sore that he wouldn't even take a popisicle.
Just try to keep her comfortable with pain meds and lots of popsicles and she should be okay.
Good luck -