How is this different than if he were sick from the flu? If he needs to put ice on his nether regions, does he need to let the kids see that? I see no point in that. He should have a blanket on his lap or go rest in your bedroom and keep the door closed. Tell the kids to be quiet and let dad be alone because he isn't feeling well.
As for when you go to the doctor's office, I think you simply say "dad has a doctor appointment like when you have to go for a check up," and leave it at that. They're not going to be allowed to see anything, and I wouldn't recommend going into the exam room prior with the kids in tow. Wait in the waiting room and take him home after without discussion with the kids.
For you or dad to make a big production or over explain the whys and hows over the surgery in front of the kids is pointless and opens unnecessary cans of worms IMO. Make this business as usual and keep the whining, comments, etc. between the two of you.