"...a Remington 700 or similar sniper rifle..." A scoped Rem 700, in any calibre, is not a sniper's rifle unless it has been heavily modified. A standard scoped Rem 700 is a hunting rifle. Period.
"...just not sub-MOA..." Few semi-autos are. That'd be less than 1 inch groups at 100 yards. An AR can be turned into a very fine target rifle though. Not inexpensively, but it can be done. Heavily modified AR's are the standard National Match rifles these days.
"....223 wasn't suitable for deer..is that wrong?..." Depends on the bullet. Most factory .223 ammo is loaded with varmint bullets. Varmints bullets are designed to expand rapidly without much penetration. They're not suitable for deer sized game. Deer sized game requires bullets that penetrate then start expanding.
The size of your local deer matters too. Up here, a deer can run nearly 300 pounds live weight. That'd be a huge deer though, but a 200 pound plus deer isn't entirely unusual. Deer are smaller elsewhere.
1) There is no one calibre. However, what you intend doing with the rifle and the ammo matters. No calibre is suitable for 600 yard hunting shots if you can't hit a 9" pie plate at that distance every time. A .308 target rifle with 168 grain match grade bullets will do nicely out to 600, but you cannot just pick one up and expect to hit the bull at 600.
2) .22 Long Rifle. The least expensive centre fire ammo is likely the .223.
