I have fired an agency owned 686-3 with full house 125-grain Winchester .357 loads on a 60-round course. I had been use to firing a 2.5 inch model 66 and a 4-inch model 19 a lot. I have never weighed the 686 but felt it was slightly heavier, and the muzzle jump slightly less.
I would not use 38 Special loads for deer. I have seen deer put down in a hunting situation with 125-grain, 140-grain, and 158-grain factory loads in the .357. I have also seen a number of deer injured by cars put out of their misery with the described loads. One of the concerns when up close to deer being put down was over penetration and putting bystanders at risk.
I owned, and loaded for the .357 for years before I received my L.E. training. The .38 and 357 cases are simple to load. When loading lead bullets the goal was to keep velocity below a 1000 fps to reduce leading. I suggest getting a good manual, and go from there.