Yes, you can.
Yes, this rifle scope can be used on a rifle chambered in .338 Lapua.
The rifle scope should be just fine on a rifle chambered in .338 Lapua and will hold up to the recoil. If by chance you run into an issue, Burris has a fantastic lifetime warranty on these products.
Yes, this should allow you to make 1000 yards shots (although a 20moa rail would be recommended).
As long as the ammo box has all the necessary information, it should work with the box info.
Any location that is convenient for you - most right-handed shooters affix the remote to the left side of the stock for activation using the left thumb.
From the user manual:
The cable can be attached to either side of the Eliminator. The cable is a flexible coil, and has a length of 14-24 inches to accomodate various mounting positions.
Mark H.
It will mount to either a Weaver or Picatinny style mounting base.