With rimfire or air power, shot placement is key. Coons are tough, and you can't shoot them straight on in the head, even .22 rimfire will bounce off their thick frontal skull! The side of the skull is relatively weak, however, so a profile shot to the brainpan will put them immediately down.
Skunks are different because of their spray factor, which WILL leak out no matter how you shoot them. It can be done scent free, however, it's just a matter of shot placement time. If you don't kill them immediately, it will come out, even a brain shot will allow them to leak their spray out relatively quickly. To do the job odor free, you need to dig a deep hole beforehand, put eggs or tuna or whatever bait you use near the hole. Let them come and eat for free for a few nights to get used to the feeding routine and comfortable with the set up, then use a double-lung shot (profile only) to put them down.* Puncturing both lungs makes them incapable of spraying, and even delays their spray from leaking for a minute. It will eventually leak out, but you have time to bury them before it does. If you only get one lung, however, you will definitely smell them.
I've done my pest control with .22 cal. air rifles. I don't think I'd want anything less for coons and skunks because they're getting up to the limit of spring-piston quarry. I use an Air Arms Pro-Elite, which is no longer made and out of your price range. You may be able to find a Beeman R-9 (used) or RWS equivalent to meet your price point.
With air guns, it's really about precision and shot placement. Don't believe the overly-hyped marketing velocity numbers. I've never see any that come close to what a real-world chronograph will measure with real-world ammo. With pellets, there is no such thing as "energy dump", you only have caliber (cross section) and penetration. But if you hit the right mark, that's all you need.
3 foot pounds to a squirrel's head and it's lights out for the squirrel. That's an amazingly small amount of energy, but directed to the profile or back of the head, it's all you need.
*It helps to study anatomy to get the shot placement right. Look at skeletons if you have a nearby natural museum, or see if you can get 3-D models online. Rotate the model around and get a good idea of brain, heart and lung placement from all angles. Carefully consider what angles will give you a good shot, and what angles should be avoided. Most people tend to shoot forward of the lungs, towards the shoulders. They're located a bit further back than you might expect. Look and the profile and consider how much of a "true profile" you will need to have a certain double lung shot.
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