My daughter had her tonsils/adnoids out when she was 2-1/2. She had sleep apnea and ate minimally as her tonsils were so enlarged that it was difficult for her to eat. It was scary to listen to her stop breathing as she slept.
We did NOT have a sleep study - the other symptoms were there...why put her through it?
She had a WONDERFUL pediatric ENT who scheduled her surgery at a hospital where there was a pediatric anesthesiologist. (Not all hospitals have them - you might want to check it out.) It's normally done as an outpatient procedure, but because she was young, they kept her overnight. We felt safer that way.
The change in her breathing and eating occurred within 24 hours post-op. It was amazing! Well worth it. Day 1, you would have never known she had surgery. However, days 2 and 3 were pretty rough for her. They told us this was normal.
Poor impulse control is VERY normal for a 3 year old boy. Most grow out of it. I wouldn't worry about it for a few years. Those who don't outgrow it have other options as they grow older.