First thing I would recommend is not shoot the bird with a choke and load that is going to blow up the head and neck (especially if you want to use the bird's actual head for your mount). Shooting a bird in the head and neck with a tight choke and maximum load will result in damage to the neck area below the head that is difficult to repair. Similarly, a head shot with same will ruin the head for use in your mount. However, heads are interchangeable, so if you ruin your birds head, it can be replaced easily enough,...it just would not be YOUR birds head. (Capes can be replaced, as well, so if you were to kill a gobbler with special characteristics,...such as multiple beards or fantastic spurs,...you could have a mount done using your beards/spurs but with a replacement cape if you wanted to go that route)
Secondly, as others have stated, if your bird is flopping after the shot, try to get to him as quickly as possible and either pin him down to the ground solidly, or grab the feet and hang on for dear life. If the bird is not flopping, don't touch him. Grabbing an otherwise motionless gobbler will very often set off reflexive flopping. Just be ready to grab a motionless bird if he starts flopping (that can happen for five minutes or more after a bird is down so don't pick him up too quickly and be ready,...again, pinning him down or grabbing the feet and holding him up).
Pretty much every bird is going to lose some feathers in this process. Some feather loss should be expected and losing small amounts of the less important feathers is no big deal. However, pick up any tail feathers or primary wing feathers to take with you. Most (if not all) taxidermists that do turkeys will have a supply of replacement feathers on hand to do repairs.
Minimizing blood on the feathers is a good idea, but it is not a deal breaker. The taxidermy process involves a thorough washing and drying of the cape. That process will remove the blood from the cape.
Between the shot and the taxidermist, just treat your bird gently. Dragging him through the mud or briars will likely do damage that will be hard for even the best taxidermist to deal with. The more of that that occurs before you get him to the taxidermist, the harder he will be to make look good. Just be as careful with the bird as you can and you should be fine. Don't throw him in the back of your truck and drive around with him rolling about for a few hours,...just use common sense and protect him.
If you are not going to be able to get him to a taxidermist right away (which is generally the case), if at all possible freeze your bird. Place him in a trash bag or two, being careful with the feathers while doing so, and put him in a freezer. If you cannot freeze your bird, you must keep him cold enough so that the cape does not start slipping and the feathers start falling out. If that happens, you are screwed. Depending on how cold you can keep a bird (without freezing), it should be delivered to your taxidermist within a few days. If you can freeze it, it will keep for a few months (or longer),...which should be plenty of time to make your decision on who to use for mounting the bird.
If you have to ship the bird to your taxidermist, make sure it is placed in waterproof packaging (sealed, heavy trash bags or equal) frozen solid when you ship, and make sure it is packed such that it will be protected and not jostle around in the box, and then ship by a method that will get the bird to your guy in no more than a day or two. If you ship a bird that, for whatever reason, starts to thaw and the box starts leaking, the shipper will discard it and you will be out of luck.
Finally, one of the ways to make all of this easier is to learn to cape your birds yourself. Dealing with a cape, both in freezing and shipping, is much easier than dealing with an entire bird, plus you get to eat the meat. Caping is a pretty simple process. Anybody can do it.