Whatever I have at the time. Bears are a problem at the ranch where I work. I have used everything from .243 Winchester to .338Winchester Magnum. This year the .280 Remington, the .25-06 and the .300 Win Mag scored. The .280 Remington scored on one coyote, and also my deer, a 3 x 3 Muley. Left the front half at the ranch for the boys, and the hind weighed 79 pounds to the locker.
My SOP is to always take two guns. My choices depend largely on the terrain. A couple of the places I hunt are pretty brushy, so I'll take the 870 super express and the .45. If I'm heading to a more open place it's the M96Swede .22-250 and the .45. I have yet to get a shot at one with the .45. Hopefully this season. Nothing against the AR platform...I just don't have the desire or the dough to own one. I like my bolts.
When I go out with a partner we flip for who takes the rifle and who takes the shotgun.
I would love to use an AR for coyotes, but I'm too late to legally have one in California, thanks to the misinformed legislators who passed the anti-gun laws covering such firearms as the AR-15 and related so-called "Assault Weapons," which by-the-way seems to be a term that I have yet to see defined.
Instead, I use either a .270 bolt action for coyotes and hogs, or a .22-250 Remington (40X bench-rest single shot), a Cooper .20 VarTarg (Love the rifle for ground squirrels, but haven't tried it for coyotes....yet). 55 grain Noslers or V-Maxes for the .22-250, and 32 grain Noslers for the 20-caliber Cooper. I don't know if the 32 grainers will suffice for coyotes, and may want to step up to 40 grain fodder.
I haven't really paid much attention to shooting dogs with the Cooper because I am so used to the other two rifles, and as such, I haven't loaded any 40 grain bullets for it and have no idea if I should load up some cartridges with 40 grain bullets to be used only for coyotes. I do use the 32 grainers for ground squirrels, however, but only in the Cooper. The .22-250 is really too heavy a caliber for ground squirrels IMO, unless shooting at ranges of several hundred yards. Most of my ground squirrel shots, however, present at far closer ranges, hence the lighter cartridges. All my center fire rifles use my own reloads.
There are many cal legal ar-15 formats out there. As long as the rifle has a 10 round mag, and the mag release requires a bullet button or tool. Other than that your legal to own.
I use a .243 NEF Handi Rifle and a Mossberg 835 with a 20 inch barrel and thumbhole stock. They are both nice and short which I like for walking and if I climb into one of my deer stands and call I can pull either one up with me easy enough. Someday I'd like to get an AR but the fact that you can get 3 bolts for the cost really makes them tough to justify.