Generally my dad and Iwill gut our deer near to where they fall. If it is right on the bait pile I will drag it off in the woods a few yards. What I like to do is find a little ridge or hump to lay the deer on, so it's hind quarters are lower than the chest. Cleaning a deer on the flat level is OK too. I start by slitting up the abdomen, with one finger on either side of the knife blade to keep the intestines away. I will slit the hide right up to the neck. My dad only slits his deer up to the bottom of the sternum. What I have found is that it doesn't take much pressure to push a good sharp knife all the way through the sternum up to the neck. My dad ends up up to his elbows in deer trying to blindly cut the windpipe. when a deer is split all the way up I can just grab the windpipe through the chest and cut it off. I don't even get blood past my wrists. We both "tailpipe" our deer. We don't split the pelvis but instead carefully slice around the rectum and genitalia and pull the whole works out from the inside. We then drag our deer back to the truck.
Once we have the deer home or to camp, we will rinse the cavity out with a few buckets of cold water, then hang the deer from the neck and let it cool. It is usually bordering on freezing during deer season up here, so we skin and butcher the deer within a week or two. If it is really warm we will skin and quarter it the next day and freeze the quarters to work on later.
We leave the gut piles in the woods for coyotes. Deer see their own kind dead or half eaten all the time along the side of the road, I think. I tend to think the little ones aren't as afriad, but I wonder if the big deer don't skip town when they find a gutpile.