Tombread: Don't know how much night-hunting you've done, so I may be preaching to the choir.
I use a red lens-cover on my QBeam. I hold it so that as I sweep, and eyes appear, I can put only the edge of the light on the eyes. The animals react less to the light that way, and will continue on in to a call. I'll also move the light completely off the animal for intervals...Or, two flashlights, such as a Streamlight for the initial finding, and a much weaker light to merely light up the eyes.
At night, for whatever reason, it takes several shots at different animals before the action stops.
I'd expect most varmints to come in to within 50 yards, to a call, so you can pick your cartridge in part on those terms. Of course, 200-yard shots in daylight are marginal on a coyote, if you're using a .22WMR in a package cost of around $300.
A bolt-action .223 works well, and the ammo has been cheap. If you start reloading, it's even cheaper. I see no need for the exotic bullets, since the GI stuff works well on varmints. Since there's no recoil, most any brand of scope will work--with the caveat from other threads that you must check to see if the point of aim is different from, say, three-power when at nine-power.
Most any good used rifle will work. You can go thru the tfl archives on re-bedding. You can add a custom trigger later, if necessary. (I didn't realize when I bought it that my new Ruger 77 in .223 had a "tort-liability" trigger. Mr. Timney's unit took a tad of grinding for the safety to work, but it was a piece of cake for the installation.)
And so on...
Best luck, Art