Deer have visited me a couple times this year. First clue was deer tracks at the set obviously. In regards to the anatomy of a deers foot, I'm going to compare it to an 1 1/2" pipe, round and smooth. If you were to set a trap off with a pipe, you will pull the trap out of the bed, reach the end of the anchor chain and then pull the pipe right out of the jaws of the trap. If you get a good pad catch on a canine, that canine is not going to pull out of that trap as long as it has good springs and is in good working order. There are far too many contours of soft tissue in the coyote/fox foot to simply pull out. A couple times my traps have held two of the toe pads and the coyote was jumping around as if he was on a pogo stick. There has been mention of a catch circle, never once has there been a "spin" at the base of the chain when a deer has set off a trap. It has always been extended straight to one direction and very little disturbance at the set. As far as hair...what length are you ending up with in the jaws? Deer have very short leg hair with my experience. By my comparison, coyotes have longer foot fur than deer do on their legs. In my area color of fur doesn't mean anything, coyotes come in brown to red to grey and everything in between. I have never knowingly ran into the problem of "rolling" on a set, but if that were the case I would imagine that the hair in the jaws would be long and not short. If you are using 2 to 3 pounds of pan tension I would think that if a coyote set off that trap under good circumstances that it would be buried up to the pad. If you throw in frozen ground or wet ground with thick soil then the scenario changes...that's another topic and/or headache.
Sometimes this trapping thing can get frustrating with what Mother Nature gives you but that is the game that's being played. I am defiantly here to tell you that failure in this game is the best teacher. You are not the only one that has experienced these problems, you can read about it in these threads for hours, I have!! Just keep plugging away, things will work out.