HoosierD, The M14 type rifle with either an 18" or 22" barrel with a 1:10 ROT will do everything you have mentioned.
Generally: 147/150 gr. inside 400 yards, 168 gr. 350 to 800, and LC M118 LR or equivalent 750 to 1000 yards.
You can reach 1000 yards with the 18", but the 22" is going to provided you with an increase in muzzle velocity.
PowderMan mentioned the SAGE EBR chassis, and yes it is an excellent way to enhance the accuracy of the M14 type rifle...
Google the ARMY M14EBR-RI and check out the accuracy they get out of rack grade USGI M14s bolted into the EBR chassis.
This is my long range M14, custom built for me by
Smith Enterprise (SEI).
It is a Crazy Horse M21A5 with a 22" medium heavy barrel with a 1:10 ROT.
The optional Wind Talker suppressor helps to further tighten up groups
I owned a really nice AR-10 with a 20" medium weight barrel, plus a
complete 16" carbine upper... I sold it to fund another SEI M14 build.