22mag will work for your application. Its limit is about 125 yards on a coyote.Its really not enough gun for dogs though. Its just right for the bobcat if you don't want to tear it up. However, 22 mag is marginal your gonna get wounded dogs and runners.
22-250 is way more gun then you need and costs more to shoot. If you lived in the west then yeah 22-250.
.223 is probably your best option. However there are more options to consider if you want to be different but since its hard to find ammo and expensive for less popular loads maybe not so good. some of the others to look at are the 22 hornet and .222 rem. Both these rounds can be reloaded down to 22 mag velocity for your bobcats if you really wanted.
you can pick up a 22 mag for cheap! the centerfire guns run a little more. you could get a savage 22 mag for like 220 bucks and it will probably be a tack driver. the cz is about 400. sako quad 500.
centerfire you could get the stevens for like 300. or cz and most other brands your looking at approx 600-1,000.
you could get a stevens and savage for like 500 and have both !
I personaly would rather save up and get a cz no matter what the caliber. Sako if your rich.
definently don't discount the savage 24f you can get 20 or 12 guage and an assortment of rifle calibers to pick from some of which can be loaded to 22mag speeds or run em hot its up to you.baikal also makes combo guns with 2 triggers which will allow for faster barrel selection.
soooo many choices but variety is the spice of life. ! take your time, read all you can , shoulder all you can.