If you were starting from scratch with a blank canvas and were designing your ultimate long range varmint rig for coyote/red fox (Aust). What Calibre would you choose and why? It would need to be an effective one shot killer from 100yds to at least 750yds or greater. Please go into detail with barrel length, twist rate, stock setup and if it can be done accurately enough with a factory rig or would I require a fully customized rifle. Then suggest optically a power range that would be suitable to see these small critters at extended range along with reticle choice. I'm not asking for brand names as I think we all have our preferences, just the technical requirements to make an extremely accurate and deadly varminter.
That's a hand full of a request.
If you want to keep (sell) the fur, then you are out of luck, cuz anything that will kill a 'yote at 750, will absolutely wreck them at 500yds, and turn them to mush pudding at 250.
But if you are shooting them, without concern for the pelts, there IS hope ;)
Any of the cartridges in the energy/velocity range of the 243, 6mm Rem, 260 Rem 264-WM families, and equivalent wildkitties will do what you want.
Muzzle velocity with medium to heavy bullets should be around 3200-ish or faster, and more faster is more better.
It should be a caliber that V-Maxs and Blitzkings are available - I tried long range feral dogs with a 300 WM, and I could hit them, but the bullets kept going... and going and going and going.
This limits you to 243 and 264 bore size. With a very few extreme exceptions like the 8" twist 22-250 and Swift, this eliminates the .224 bore size.
For a barrel, I would have 26" as a minimum, with 28+ being preferable, and a 3/4" or larger muzzle .
For a scope, generally a 6x20 to 8x32 is common place for long range varminting, but a 4.5 to 14 is fine. A good fixed 10x will do fine on these critters 99% of the time. DON'T cheap out on the scope. The scope should be from the better brands, and the top of their line.
What is VERY important for a rifle like this, is elevation repeatability... when you range a 'yote at 800 yds, and dial the elevation turret, it MUST go to the same place each time... and there are many scopes that won't do this simple chore.
The rifle should be built on a Rem 700, a Win M-70, or any of the better custom actions (BAT, etc).
The stock should have a wide fore end, aka Varmint stock. Some of the best are the H-S Precision stocks. It should be glass bedded.
Weight should be roughly 12 to 15 pounds, all up and ready to shoot.
It should be fit for a Harris bipod, cuz you will do most of your shooting from one.
There are factory rifles that will do this. Remington Varmints in 243 and 6mm Rem. Remington Sendero's in 264. Sako and Tikka equivalents are very fine rifles.
Look to lighten your wallet from $2k+ for a factory rig, to twice that (plus) for a custom rifle..
And you will need to hand load, cuz factory ammo will rarely do the kind of accuracy needed for long range shooting.
But, at these distances, the rifle is only 50% of the battle. The rest is up to you.