Re: whats the best 22 automatic rifle?
I've owned a number of .22 semi autos. My two favorites were the Remington Nylon 66 (no longer in production) and the Remington 597. I like the 597 slightly better for its styling and because it's available in stainless steel. The Ruger 10/22 is nearly as good and can be customized like crazy. But customizing does not seem to be your goal.
A coyote would seem to be at the upper end of the animal size scale for a .22 long rifle. That round is more suited to squirrels and the like. But at moderate ranges and with good shot placement, a .22 long rifle can be effective on coyotes. Just be ready for a follow up shot in the event the first shot hits but does not kill the coyote. Besides the ethical issues of wounding animals, there's the practical issue of dealing with a wounded predator. A .22 magnum might be a better choice. Or a .22 centerfire rifle.
And if the gun is going to be lying around a lot and not cleaned after every shooting session, then I'd recommend a stainless steel gun over a carbon steel gun. The stainless gun will hold up much better.