Funny...I just observed a tonsillectomy and adenoidectomy today. Depending on the speed (and age/experience) of your ENT surgeon, the actual procedure could take anywhere from 20-40 minutes (closer to 20). With the anesthesia and prep time, they usually block an hour or sometimes two depending on your hospital.
Generally, you are put to sleep and a breathing tube is placed into your trachea (the breathing pipe) and a machine is used to assist your breathing. This is done to keep blood and other secretions from seeping into your lungs during the procedure. Then, the adenoids and tonsils are surgically removed. This part is really quick. The surgeon then looks around and makes sure that there is no bleeding as these are VERY vascular areas. Once he/she is convinced that there is no bleeding, you're woken up a bit, the breathing tube is removed once you're breathing on your own (the drugs are amnestic and will keep you from remembering this), the breathing tube is taken out and you'll be taken to the recovery room.
In the recovery room (now known as PACU, Post Anesthesia Care Unit), you are allowed to fully recover from the anesthesia. You are given pain medicine (typically intravenous narcotics like morphine) to help you out. Once you are able to tolerate something to drink, you'll be free to go home. You generally get a prescription for an oral narcotic (usually Tylenol with codeine or percocet or one of their derivatives) to relieve the post-op pain.
The largest complications are nausea/vomiting after the procedure and bleeding from the surgical site.
As far as returning to flying, you'll have to talk to your family doctor or ENT surgeon, but I would NOT recommend flying until 24 hours after you've taken your last dose of your oral narcotic. In addition, if you're still not feeling 100% 24 hours after your last dose of oral pain meds, I wouldn't fly until you did.
Hope this helps. Things may be different from how I've described because I am an Emergency Medicine doc, not an ENT. As always, you should really direct these types of questions to your surgeon and/or Anesthesiologist.
You land a million planes safely, then you have one little mid-air and you never hear the end of it - Pushing Tin