J_Darnley: Why the heck don't the doctors do surgery at night? They sit on their expensive hands all god damn night and let the patients collapse and die.
One thing that seems to help is to designate at least the main surgeon, the anesthesiologist, and the surgical nurse as only performing those roles (with the check boxes at the bottom left of the character sheet). Then you only need any other doctor and any other nurse. You may then want to uncheck those boxes for others or the specialists could be sitting around doing nothing while some other doctor who can do anything does the surgery. If you don't limit roles then enough doctors or nurses may stay busy enough that not everyone is available at the same time.
Also, it sounds like the qualifications of the doctors and nurses affect the speed in some ways and I'm not sure exactly how that works but it might be responsible in some cases.
J_Darnley: Why the heck doesn't the orthapaedic surgery perform the operations it needs to? Some guy needs a hip replacement or some other surgery but allegedly the department can't do it. I think I've got all the things needed in the tooltip but I still can't send them to this "advanced" surgery.
Some surgery requires HDU hospitalization and I think the recent patches are better at explaining this. It might not be available if you don't have the HDU bed or it might be possible to click the surgery but nothing will happen unless you clicked HDU first (I think I've seen both). At least with the most recent version sometimes things are just not available from the ICU when they are available from patients in the department; I'm fairly sure this is just a bug. Sometimes the doctor can select the right thing but you can't so changing to doctor control can fix some issues. Sometimes doctors can transfer patients out of the ICU, although I don't know if they will ever do that before surgery. Sometimes there are synchronization issues if you have had the character sheet of the patient open while something changed (at least with department changes) and changing to a different character sheet and back will make more options available.
I haven't seen it with surgery so far but I did see what I'm fairly sure was a race condition involving scheduling an x-ray, unscheduling it, then transfering the patient out of the hospital and after that no more x-rays would be performed even though staff was idle (even after firing/hiring and shift changes). I think it might have been unscheduling it at just the right moment that caused the issue. I've seen cases of technicians getting more obviously stuck in labs and sonography units and in those cases firing them and waiting until after midnight sometimes helps, although IIRC sometimes it then gets into the "idle staff won't do anything" state. My memory isn't that great but I think I saw an office doctor get obviously stuck once as well but firing and waiting until after midnight seemed to fix the problem. I think I've had one or more issues along those lines in every game I've played that I didn't run out of money quickly.
I'm not certain but I think the "depressed" condition might cause staff to sometimes stay idle and not do anything for a shorter period of time when they could be doing something. If so, if one or more particular doctor or nurse needed for surgery is depressed it could be extra hard to get started.