I agree on all pointsA couple thoughts: I've dressed deer both ways and I prefer to start at the anus and work my way to the chest. That's just me. Taking care of the but-hole first allows me to do that without worrying about the mess at the other end. Once I cut the skin around it, I then run my knife all around it, inside the pelvis.
When dressing a gut-shot deer(and eventually every hunter will), GO SLOW. I have done my share and in 99% of the cases, I dare you to tell me it was gut-shot.
As for washing out the carcass. If you want to wash out a carcass, do it with dry rags. I would not recommend using water unless you are in the butchering phase meaning, the wet meat is about to go into the freezer. Getting any carcass wet and then letting is sit or hang for any time, you run the risk of promoting bacterial growth, especially in a gut-shot deer.
.....and a quick note in regards to slicing them open. Do NOT cut into the ribcage if you shoot a good deer that you want a shoulder mount done on. Stop your incision before you hit the sternum so as not to cut into the portion of the deer that the taxidermist woin't want cut up. I cut all my deer that way now, it's not necessary to slice up through the rib cage, just reach in, separate the windpipe and remove the heart/lungs.