YEP!!! For antelope for sure!!!

Each species you mentioned ...at least for me requires a different gun and in some cases several.
Rabbits are a rimfire thing if you plan to eat them...otherwise anything will work.

Coyotes need something fast and handy in a 22 or 24 caliber. I need at least two rifles just for this purpose and probably more.

Prairie dogs require at least 6 rifles and depending on how many friends you have probably more. I take at least 4 everytrip. Anything from a 17 rimfire to my .300 Win mag will work but I prfer something I can shoot all day.

Antelope, just to satify DOW requirements it'll have to be at least a .24 cal and 80 gr. bullets. Need a few of these too. I take two every year.

You didn't even ask about deer and elk.

My advise is to not try to cover too many bases with one gun. I like "target specific" guns myself.

I wish I could afford buying one of everything but right now the budget isn't there for it. I already have a 7mm Mauser that I use for deer but it's a little to spendy for me to spend on antelope. It really isn't all that much to shoot but its getting a little tougher each year to find shells for it.

If you're concerned you're going to run out of ammo somehow for the 7x57, then you've probably just made the case for the .243.

You could probably find .243 ammo at an ice cream store, it's that common. It's THE perfect antelope round, and I've used it on deer for years and have never lost an animal. It knocks the tar out of coyotes. For years I had but two calibers in the closet, a .243 and a .30-06. With just the two of them, I covered everything from jackrabbits to moose. I've taken two elk with a .243, but that's probably the caliber's reasonable limit - stick to deer and below.

Just 4? If there was ever a varmint designed to empty out your gunsafe for the day, it's the prairie dog.