I can't vote for 2 so here are the best 2 imo. The .17HMR and the .22WMR will serve you best from the choices you have listed. The .17HM2 or Mach2 will reach out to those distances but I doubt its effectiveness against coyotes. Same with .22lr (at longer ranges anyway).
The .22lr will do what you want except for the coyote. For that I would suggest a .22-250 which will reach to 300 yards. I don't have that much experience with the .22mag to comment but I would suspect it might be good to 150 yards but I'd question it's drop dead ability at that range. However the .22lr can stop coyote at ranges of 50 yards or less. So it all depends on the range your looking to take a coyote. Also a .22mag would be a bit much for a squirrel or rabbit. I think the best round would be the .22lr.
"A free people ought not only to be armed and disciplined, but they should have sufficient arms and ammunition to maintain a status of independence from any who might attempt to abuse them, which would include their own government." -- Unknown author -- Falsely attributed to George Washington -- Patriot Guard Rider
might consider an ar15 in 223/5.56 with a cmmg conversion to 22lr or with a dedicated 22lr upper to go with the centerfire then your really versitile.
"Gun control: The theory that a woman found dead in an alley, raped and strangled with her panty hose, is somehow morally superior to a woman explaining to police how her attacker got that fatal bullet wound." L. Neil Smith
The problem with being stupid is you cannot simply decide to stop doing dumb things...