Have you thought about the 260 Remington or 6.5 Creedmoor? All 3 calibers are almost identical on performance across the board.
I shoot both the 260 and 6.5 CM,and enjoy both calibers.The thing I really like about the Creedmoor is you can really take advantage of the longer match grade bullets,and still remain well within the limits of magazine length's of bolt action rifles,and still have the velocity of the 260 or 6.5x47.
My 260 is an AR10,so I must load it to magazine length,but it is used as a hunting rifle,so that doesn't affect many concerns on maximum performance like I put on the bolt action 6.5 CM.
As far as your question's on what barrel & action to use- Do you want to build it,or have it built by a gunsmith?
If you want to do it,a Savage action would be the cheapest/easiest to work with. I used a Savage Varmint action for my 6.5 CM build,it's a standard stainless action with the 1.5lb Target action trigger. Or you could use almost any commercial action or custom action,but most would need to be assembled by a gunsmith if you don't have the know how and tool's to do it.
You can go mild to wild on a custom barrels,it really depends on how much you want to spend,and just how long you want to wait for one.
E.R.Shaw,Shilen,McGowen are great lower priced barrels,Hart,Kreiger,Lilja,Bartlein are some of the better premium barrel makers.