im not quite sure about the .22  but a .22 mag would definately do the job. I wouldn't be aiming for the vitals tho, it will be a very slow death for the animal. Hit it in the head, any .22 will do and it will drop dead right before your eyes. No suffering no tracking.

The problem with the .22 to the vitals is that it will probably die but you will also probably NEVER find it. Such a small hole probably wouldnt bleed or do as much damage as you'd like. Get out that 12 guage that you were talkin about before and put a slug in it from 50 yrs out. That'd be much easier and more humane.

So to answer ur questions, prolly wont kill a coyote to the vitals very easily but one to the head will do it just fine.

.22 hornet.
thats all i've got to say

oh lol i didnt catch the vibe that it you were gonna use the pelt

then if all you have is a .22.... i'd ONLY take a head shot

i wouldnt even try for a vital shot, even if you make the perfect shot it wont go down for a long time and you'll probably never find it

If you want the pelt snare him.

buger, i don't mean to critical or disagree with you methods, but there aren't any parts to snare wires, and it costs maybe two dollars and change for 50 feet of 24 gauge steel wire, which is all you need for perhaps hundreds of snares.  also, as long as the wire is not broken by a larger animal, or lost because it came off of the anchor when the animal panicked (the latter is preventible most of the time by using strong anchors and connections, but there is always that freak accident)

about opposums, they could pose a problem, but with all of the snares you can set, i wouldn't worry about them.

and out of order response: it only takes a minute to set a wire snare.  here is the proccess:
1. before you go out, twist a small loop into the end of the wire length that you cut.  it doesn't need to be much bigger than a milimeter or two larger around than the wire it is made of. 
2. cut out a one and a half to two foot stake for use as an anchor(make it longer and stronger depending on the animal you want to catch) (not required.  you can use saplings and other natural anchors instead)
3. now head out to the path way of the desired catch(paw prints help figure out what is traveling this path)
and either place your anchor next to the path, or pick a natural one next to the path
4. thread the end of the wire through the little loop you made earlier to make a noose.
5. attach the free end of the wire to the anchor so that it will not come off
6. position the noose so that it is at about head hight of the animal you want to catch, making sure that the animal's head will fit through it without trouble
7. i have read instructions that you are supposed to rub the wire with the inside of a peice of bark cut from a sapling to remove your smell, but i don't think that it is really neccesary.

that is about it.  not very complex or anything.

yes, usually

you can also probably find it in some section or another of walmart.

the craft section has copper wire, but that might not be strong enough.  i am not sure.

I ussualy use copper wire I salvage from lamp cords.
I know Im cheap.