I have been looking at 204 ruger or 22-250 for caliber.
Is a heavy barell needed for Varmint/Coyote hunting???
FWIW, my brother is also left handed. He uses a CZ 527 American in .204 Ruger. I have two CZ 527 rifles in .204 Ruger, the standard weight and the varmint weight. We both use 35 grain Berger bullets. The previous poster is spot on about fast handling rifles for close in calling.
Later in the year, when coyotes are more wary and come in with a bit more caution, we both switch over to a 22-250 with 22 inch tubes. His is about a number 3 taper and mine is closer to a number 4 taper. The short length still maintains some decent handling characteristics at the expense of some muzzle velocity, but the weight adds stability.
I too have shot a boat load of coyotes (well over 100) with a .204, and I think they are great out to a couple of hundred yards. If you just want a dead dog, start with a 22-250 using polymer tipped bullets and work up from there.