I have seen more foxes in my garden than mice in my kitchen (except the dead ones on their way to the micecream box).
I have lost three chickens to foxes.
So far I have not lost any chickens to mice, nor have I had a problem with mouse excrement (though treading on a slug on the tiled floor is somewhat unpleasant)
It appears to me to be as legitimate to set a trap in the areas where I know the mice run, even though that is not where they cause the problem, as it is to seek out a fox, even though at the time it is not in my garden.
As it is, I don't usually bother too much about the mice until they become an apparent issue (though I do have concerns about cables etc being chewed through) and I don't go fox hunting, I fox proof the chicken run.
If I were trapping live mice and microwaving them for fun, that would not really be very nice, would it? As it is I am not sure how long it takes a mouse to die in a spring trap - possibly seconds, possibly hours depending on how it is caught.
..d