I just started trapping myself and I've talked to alot of old timers trying to get some tips. This is what I've come away with.

Coyotes: Use snares rather than leg hold traps. Saturate an area with many snares in every travelway. Everyone I've talked to has had success doing this as opposed to leg holds

Foxes: make sets at along fence lines.

I haven't learned much about beaver trapping so no dice there.

hope this little bit helps.