It's never clearly stated above, but the one thing you really shouldn't do is talk to the dog (e.g., say "here" or "heel") after you signal for the birds. That is against the AKC rules. The judges will have to assume you are steadying the dog (regardless of your real intent) and will score it as a controlled break. At Senior a controlled break will cost you a bunch of trainability points. At Master you just failed the test before the marks even went down. If you start using hand gestures or pat your leg the judges will probably look at it the same as a controlled break.
Despite the sound effects in the field, plenty of dogs fail to see marks at AKC tests. IMHO some of the AKC judges are not as keyed in to the visibility of their marks as the HRC judges tend to be. Also, the popper guns in the field can sometimes create echoes from places that confuse dogs if they are not properly aimed. Can you tell these are pet peaves of mine?
I've been pretty successful in having my dogs see marks. There are plenty of things that can be done while staying within the rules. I'll let you in on some things I do. I didn't invent any of what I am going to share here. I have judged a bunch of AKC tests and I have seen plenty of handlers doing the things I am going to tell you. I haven't run any HRC tests in a while, but when I did I used to see handlers there doing many of these same things. Keep in mind though, I am right handed and I only train my dogs to heel on the left side. If you are left-handed or you have a two-sided heeling dog you will have to modify things accordingly.
Before I run I try to get up very close to the line to see things from as close as possible to the working dog's vantage point. You can head off lots of trouble by simply figuring out how to avoid placing a short dog where its sight will be blocked by something tall. You can usually get this view by carefully positioning yourself when the test dog is running or with the permission of the judges at a re-birding break.
I plan my walk from the final holding blind to the line in a way that has the dog walking toward (and looking straight at) the station that will be throwing most critical mark as we approach the line. Usually this will be the toughest memory mark. This may mean going out a different side of the holding blind than might seem most natural or picking my way around some obstacles on the way up to the line.
Once we get to the line, I will try to place the dog very tight to the place I will be standing or sitting. I generally endeavor to get the dog sitting with its spine lined up with an imaginary point that is about one-third of the way from the right-hand bird station to the middle station (in the case of a triple) or to the left-hand station (in the case of a double). Once the dog is sitting in the proper alignment I tell the dog to stay and I will go get the gun. Hopefully, the dog is looking around and checking out things in the field while I am off getting the gun. Once I am back in place I will get in as tight as possible to the dog. If I can sit on a bucket I do that. Alternatively, most judges will allow you to kneel down. That is my next preference. You can really help the dog a lot more by getting down closer to their level
If the first mark is thrown from station on the left, then once I signal for the birds I will lean forward a bit, raise the gun, and swing it to the left to point at the bird. This will put my left elbow right in front of the dog's face. The dog will naturally turn left and see the left-hand bird. I use track the bird through its arc and freeze on the point where it hits the ground. I wait there until I hear the bird boy at the next station start to call.
Then I use my right foot to pull my entire body around to point at the next station. As I rotate to the right, I am dragging my left toe in the dirt for a few inches. It makes a subtle noise that a dog will learn to react to after just a few practice sessions. Again, I track the fall all the way to the ground and freeze. If there is another mark further to the right I repeat the process again.
If the first mark is on the right, I still set up the dog with its spine aligned to a point one third of the way from the right-hand station to the next station to the left. This will generally encourage the dog to look in a direction that will allow it to at very least pick up the motion of the bird coming from the right-hand station by using its peripheral vision. Again, I raise the gun and track the bird through its arc all the way to the ground where I freeze in place. When I hear the next bird boy start to call I lean forward just a bit and use my left foot to rotate body and gun left to the next station, as so forth.
I start working with my dogs on these skills in the yard as follows. I stand next to the dog and hold one bumper in each hand, holding on to the bumper, not the cord. I then practice the foot and body motions I described above. When the dog turns its head to look in the correct direction I toss the bumper from that side to reward that behavior. Now a critical step - you need to freeze and make the dog stare at the bumper for a while before you send it for the retrieve. Otherwise, the dog will turn its head in anticipation of another mark at the slightest hint of movement, real or imagined. If the dog turns away from the bumper you threw, use body language to get the dog looking in the correct direction and then send. After a few days of this you can add the bucket and the gun. I practice with an old single shot 20. Dogs love this drill and they learn to pick up on your movements very quickly. This is a great low pressure way to end a yard session with a young dog.
You asked whether it is desirable to have a dog that pivots in place or one that repositions itself with you to see each mark. I much prefer to train my dogs to plant their butt and not move until I say so. In my experience, the dogs that get to move around tend to get happy feet and start to get into creeping problems. If you work with them enough they will twist themselves like a pretzel to see marks if you use the appropriate body language to communicate that is what they need to do.
At most AKC tests you will handle a fake shotgun. Sometimes you end up with a real gun that is unloaded. I have seen handlers cycle the action of a pump shotgun or dry fire a break open type shotgun. Some judges might have a problem with that. I don't have a problem with that when I judge, because I always verify the gun is truly unloaded and I fix it so it can't readily be loaded during the test. I have even heard a few sly handlers gently tap their ring on the barrel of the gun as a cue for the dog to turn its head.