The only TEEs I had were after the surgery, so can't help with that.
I worked in managed care for years and always double checked with the insurance company to make sure the doctor and the hospital had done the pre-auth. I also asked questions like, "Are the anesthesiologists, radiologists, and pathologists on my plan?" Then I called their offices to verify. I learned from experience with a baby who went to NICU and had a Neonatologist that you can get hit with a big bill from a doctor that's not on your plan. I also had this happen with an ER doctor who was at a participating hospital, but the ER group wasn't participating. The problem is that you don't have a choice with hospital-based physicians, and they have a captive audience, so they can do whatever they want. That is handled differently in different states and by different plans, but better to know ahead of time.
My plan paid non-participating hospital-based physicians as if they were a participating physician, but then the non-participating physician can bill the patient for the difference in the "contracted rate" and their actual charges. Some will give a discount off actual, but they don't always. For the Neonatologist, they billed $950 and my insurance company paid $200. I received a bill for $750. After many phone calls, rants, and raves, the insurance paid the remainder of the bill.