One shot is generally all you will need hunting with a handgun. If you need a second, it will likely only be a shot after they are down and trying to get up as opposed to hauling butt for the next county because you missed them with your first shot. So, makes no difference between single action or double action revolver choice.
My thoughts on the subject.... 44 mag is a good choice. It is not too big and not too small for the intended quarry. I would be somewhat selective with my shots with a black bear with a 44 mag, but for me, shots at Black Bear are not common and you take what nature serves you. Hence, I would use the 44 mag as a backup to a rifle unless you really don't care if you are successful.
Distance.... Practice. If you can hit a 6" paper plate pretty consistantly out to 100 yds, then you are good to go with the gun and caliber. If not, keep practicing or shorten your distance to where you can pretty consistantly make hits where you point them. A scope helps with consistancy and a red dot is quicker to be on target than a scope unless you practice a lot with the scoped handgun. With scope, assume you will need a rest to take anything other than a very quick shot. Hence, I carry a monopod with me when hunting deer with a scoped revolver.
I use a scoped Super Redhawk in 480 Ruger for deer hunting and would use it for bear if I was bear hunting. It is so darn large that it is very hard to carry a rifle too and actually use the handgun unless you are sitting in a stand.
My general barrel length for hunting is 5.5 -6" minimum, but longer would be better. IF I carry my 4" 41 mag revolver, it is strictly as backup and not the primary gun and shots would always be under 50 yds. I learned that the big revolvers make carrying a rifle rather uncomfortable, so for my woods hunting, I decide before I go whether I will be carrying a rifle or revolver and only take one with me unless it is a shorter barreled revolver which you can pretty easily carry on a belt holster. If you hunt with the revolver exclusively, then you accept that you can not take really long shots and hunt accordingly.
357 mag for deer... it works, but I would keep my distances well below 50 yds and choose your bullets carefully. I generally favor 41 mag or larger for whitetail hunting. Throw in bear hunting possibilites and I would go larger as my choice.
You are going to have to be honest with yourself about your true capabilities with a handgun for hunting. Only you will know at what ranges are are predicatably capable.
