E., I promise you, you are NOT being a drama queen to be worried about this. Anything that affects our children, especially surgery, is frightening, don't let anyone tell you otherwise. My son had to have major surgery to fix a blocked ureter (the tube that goes from your kidney to your bladder) just days after his 5th birthday, and even though I am a Christian and trusted God to take care of him, I was scared to death! But I am happy to report that he is a thriving, happy, healthy 16-year-old now!
Have you gotten a second opinion (from a different ENT)? I would definitely do that before going for the surgery.
I don't know if sleep apnea is different for adults than for children -- my husband has it, and uses a CPAP machine. In his case it is definitely not any kind of infection that causes it, it's a more "mechanical" problem. I suspect that's true for your daughter -- it would make sense that the enlarged tonsils and adenoids could cause some sleep issues like this.
I'm having to face a similar issue for myself -- I've had strep throat for the last nearly 3 months, and may have to have a tonsillectomy. For adults it is MUCH worse than for kids (pain and recovery-wise), so you can believe I will get a second opinion if the ENT decides I need the surgery! I wish desperately that my parents had gotten my tonsils removed when I was a child, but that was "out of favor" in the late 1960s.
A 4% chance of bleeding ... that's still a 96% chance that it won't happen!
So -- yes, get a second opinion, from a doctor who is highly respected and trusted. And then if you have to do the surgery, just trust that she will be okay and that this will be incredibly helpful for her and her quality of life.