I really don't understand the assault stat! Sometimes when I kill a non-hostile animal (deer, fox, rabbit) I get an increase in my assault stat, and sometimes I DON'T, for some reason!

Will a high assault stat have an effect on gameplay in any way (even when I have 0 in bounty)? Would a guard react if he saw me kill a fox/deer/rabbit?

Guards don't go after you if you kill a rabbit or deer.