Addressing points #1,2,4 & 5...for such a rifle I would add, as a "compulsory feature" that will help achieve the accuracy you require is it must have an adjustable trigger (can't imagine a top quality rifle not having one but would make sure just the same) and a fully free floating barrel.
#3; The game you list can be handled quite readily by any of the .25, 27, 28, 30 cal "standard, not magnum" offerings chambered these days, however the fly in the ointment for you is the 5-600 yrd range you want to shoot them at. At that range I would stick with the higher powder volume/good bullet selection, cartridges in your recoil preference such as .270 Win, 280 or 30-06. Personally at the ranges you mention I would disregard 25-06, 7-08, 308 Win, not that these cartridges can't hit at longer ranges, more that they start to loose ballistical efficiently at those ranges.
#6; this is where the rubber meets the road for 5-600 yrd shots. Rather than spending your whole wad on the rifle, glass is where I would start pricing. With the ranges you specify Huskemaw or Nightforce would fit the bill, they both offer turret range adjustments calculated to the caliber you choose. The second part ( just as important for long range shooting as the scope) of the glass formula to make use of the expensive rifle & scope would be a very good laser rangefinder that WILL range out to a 1000 yrds ( G7 comes to mind)...not a budget priced one that will only range 1/2 (my Bushnell for example) the advertised distance.
To capsulize: For the ranges you list, I would decide on the glass & rangefinder first and then fit the rifle into what's left of the budget. Everything you want in your rifle package can probably be had for 1/2 the budget you mention.