My son has had numerous procedures and trips to the hospital in his 4 years of life so it wasn't easy to get him in for the sleep study. I spent 10 minutes in the hospital parking lot just trying to get his door unlocked so that I could get him out. He was that afraid of being there. Everything ended up going well though. I had made sure to get him good and tired during the day with no naps (which meant 3 stops between home and the hospital), so by the time we checked in he was already getting groggy. We'd played up the sleep study in the week prior to going in as him getting to be a "robot" at night and that seemed to help. What little boy doesn't like robots??

In the end, I did have to hold him down some but it was over quickly and he did great at night.
Don't count on anything being able to distract your child. It may or may not work. Bring a favorite blanket, animal and video just in case but plan on having to hold him/her down at some point. It's pretty scary to have a stranger put wires, glue and tape all over your face and body. My ds doesn't tolerate stickers and hates the sight of probes let alone having them placed on his body, so we weren't able to practice prior to the study. However, I did show him pictures of other children who had the probes on so that he could see what it would look like. I assured him that it wouldn't hurt (of course he didn't believe me), but I did my best.
The T&A made a big difference for my ds as well. He had both removed at 22 months because they were so enlarged and were causing chronic tonsillitis and sinus infections. That also helped to improve his sleep for about a year until he had another surgery to repair his velum. Don't rule it out, or assume that your child will need it. Get the study over with first and go from there. One day at a time. I hope everything goes well for you. Good luck.