Buster:
Since you are not looking for something with great range, I'd suggest:
Small bore rifle
- Either the Ruger 10-22 already discussed here, or a Marlin .22. Either can be found used at reasonable prices, but the Marlin is likely to cost you a bit less.
- You could drive off or kill a dog or coyote with a .22lr, but it is unlikely to be a quick or humane kill without extremely precise bullet placement. It's fine for everything else you've mentioned, and if you're former military high expert, you should have no problem dispatching small critters at 100 yards with either, and it does not sound like you intend to shoot at longer ranges.
- Only drawbacks are that it is light on knockdown for the larger predators mentioned, and it can go right through with minimal bullet expanion at very close range.
You can likely find a good used Marlin .22 rifle for $150-$200. Marlin also used to make a less expensive line called Marlin-Glenfield, though I'm not sure if they still do. I paid $45 for a Marlin-Glenfield .22 auto quite a few years back, but you might still be able to get those for around $100 used. The Ruger 10-.22 will probably cost about $250-$350 used, depending on options, condition, scopes, etc.
The Ruger has a bit of a cumbersome magazine, but it *has* a removable magazine, so you can keep a loaded magazine ready. The Marlins generally have a tube magazine attached to the underside of the barrel, not removable, and loaded one round at a time. Not as handy for heading out to check out that noise in the night.
Small bore shotgun:
- a .410 shotgun is a good all-purpose critter control for shorter ranges. It is a very small bore shotgun. Loaded with light shot, it would chase off a dog or coyote without being likely to kill. Loaded with slug, it will easily kill either of the above ... get a over-under or pump gun, and if the birdshot does not scare them off, follow up and kill them with the slug.
Any of the old-time big names in shotguns are fine; a good used Winchester or Mossberg would be a good find, and you can probably find a Mossberg for about $125-$275 or so (used) if you look around a bit. The older Mossberg 3-shot bolt action can be had at the lower end of that price range, a decent used over/under or an old pump action at the higher end. A Winchester or Remington will run a bit more. No reaon why a Mossberg would not be up to the job, so that's probably your best low-cost solution.
A .410 is a bit more versatile, but you won't have the range of the rifle, of course. Still, you should be able to take a coyote at 30 to 50 yds with a slug if you need to. FPS drops off drastically beyond about 50 yds.
If you do get a .410 and intend to use slugs, not all manufacturers make slugs of the same diameter. Remington's are larger, and made for an open choke, and Winchester and Federal are slightly smaller, intended for a modified choke. I don't buy .410 slugs anymore, so you might want to make sure that info is still current ... but at least back when I used these, better choice of slug for choke was needed for accuracy.
As a side note, Taurus makes a revolver called the Judge that will chamber both .45 long colt and .410 shotgun shells, but they tend to run in the $300-$500 range, and if you're not that keen on handguns, the plain ol' .410 is probably better for you. I'm not a big fan of Taurus, but that sort of handgun is a utility predator control option.
The .22-250 you mentioned is one of the premier varmit cartridges around, but if you're not looking to be whacking things at 600 meters, you probably don't need that ... and if you did, you could go with .223 (5.56mm) in a good AR-15 (civilian semi-auto equiv to the M16) and do nearly as well. Sounds like you don't need to shoot at that distance, though, so I'd get the .22lr or the .410 shotgun, or one of each
