That I know of the 1885 High Wall doesn't come in .223 but for a .22 caliber it comes in .220 Swift.  For your other rifle ideas I would lean towards the Tikka.  It may be easier to get it to shoot to the ranges that you are looking at.  The Ruger is a fine rifle but it takes work to get them to be a tack driver.  Also the .224 caliber bullet really isn't that ideal for long range shooting when using the lighter bullets.  You will want to see if you can find a rifle with at least a 1-9 rate of twist for the barrel, that will stablize the heaver bullets if you want to shoot them which you can get in the Tikka and Ruger with a 1-8 in .223. 

For varmets 99% of the hunting I have done, I have done it with a Ruger #1V in .22-250 and have had fun doing it.  I topped it with a Leupold 6.5x20 AO VXIII and have no problem popping ground squirls out to 400 yards+.  Your biggest problem with a single shot is a follow up shot if needed but if you do your part it should't be. 

Thanks Critter.
That helps a bunch in making my selection. I had a friend suggest a .223 in a bolt and he knows I already have an AR. I wasn't thinking of the .223 until he mentioned it. I was in fact considering a 25/06 or a 22-250 and leaning towards the 25/06.

Your AR is a 223? I'd go for a 243 bolt action. I have the 243 bolt and the 25-06 in a bolt.

I have been thinking and a neat little round for varmets would be the old .250 Savage. 

Interested in saving hides the 25-06 or 243 will be excellent long range pred killers but will likely tear em up bad. Run into a nice bobcat you'd like to rug or mount and again might tear em up too bad. In my experience most shots are under 200 yds some out to 300 and many much closer into shotgun range. Most times I see coyotes at 500 to 600 yards its when driving not calling. With that being said I think the AR is the perfect pred gun in 90% of situations and I also like my savage predator bolt in 22-250 for when the range needs to be extended some. I really want to get my hands on the bushmaster predator AR in 223 or possibly 204 ruger in the near future. Great accuracy reports on that model.

I like my ruger #1 in .223

That was cool. Perfect for hunting.

Strongly considering a .204 Ruger in a bolt action.

I had found a Winchester single shot in .223, complete with an octegon barrel, but passed on it as I'm not real fond of the caliber. Since everything I own could conceivably be used on deer/antelope sized game, or larger, I decided to look at even smaller calibers.

The 30-06 will be relegated to bad weather/brush/truck gun usage and I'll likely reach for the .338 Win Mag or 6.5 x 284 for any game animals I care to hunt.

Biker

Still considering the lighter end of the spectrum. 

I've got a .338 Win Mag and 6.5x284 that I really like. I haven't played much with the 30-06,  outside of developing a load for it, and need to get serious about the smaller end of my battery. 

I don't hunt predators much, just targets of opportunity,  when it presents. Thinking I might hang out and wait until Winchester comes out with another run of left handed doodles and get one in .223 Remington.