What one caliber for North America?

[Faulkner]

I've taken most of the game the OP listed, with the exception of brown bears, and have done it all with a .30-06 out of a Ruger M77 MKII.

.30-06 for mountain lions and coyotes is pretty much overkill. They're more of a soft kill and I've seen both taken with .22 long rifle.

 

[c.marsh]

practice

you can use a rifle in 22lr to 450 nitro-express an if you can not place the bullet in the boiler room when it counts.pick a weapon that you can shoot well an practice a lot.i mean a lot not just 20 shots before hunting season..for long range i use the 300WM..for close up and personal you cannot beat the 45-70 marlin loaded with buffalo-bore 405 gr..

 

[Model29Lover]

I like my Browning BLR 300 Win Mag. I love the round and it will reach out to those distances without a problem whatsoever!!!! But I love a lever action, if you like bolt action get a Winchester model 70 pre '64 in 300 win mag. The 30\06 is a staple round, but at over 200 yards I would go with the 300 Win mag!!!