Yes, any of Burris's scopes can be used on an air rifle.
To mount the Burris Fullfield II 3-9x40 mm Rifle Scope to a Ruger Rifle with a picatinny base I recommend using the 1" diameter medium or low rings at the link below. The Low should work fine if your rifle doesn't have heavy barrel.
https://www.opticsplanet.com/burris-xtreme-tactical-1-two-rings.html
Low rings will typically accommodate a 40mm objective lens, medium rings will fit 44mm objective lens. However, this greatly depends on the rifle you intend to mount your scope to. Email us at Techsales@opticsplanet.com and let us know what specific rifle you have, and that way we can more accurately recommend a set of rings.
To mount the Burris Fullfield II 3-9x40 mm Rifle Scope to an AR15 platform, I would recommend using a dedicated AR15 scope mount. Typically with the AR15 platform, you want to use a high height mount to allow for a more comfortable, natural head and shooting position. I will link a few of my favorite options below that will all work for this scope.
https://www.opticsplanet.com/primary-arms-primary-arms-deluxe-extended-ar15-scope-mount-30mm.html?_iv_parent-tube-diameter=1-in&_iv_tube-diameter=1-in
https://www.opticsplanet.com/aero-precision-ultralight-scope-mount.html?_iv_parent-tube-diameter=1-in
The Fullfield II has enough tube spacing for that spec, however using a one-piece rail will give you more flexibility in mounting solutions. You do not have to stick with a two piece mounting solution on a Savage 110.
Yes, this should be compatible.
This is a non illuminated scope, and will not require a battery.
Although Burris dose have USA made optics this optic was made in china but it tested and inspected in 8th street plant in Greely Colorado.
This scope is not illuminated.
Yes, this can be mounted to an AR15, you will need a 1" ring mount like this one here: https://www.opticsplanet.com/v/padlxsmext1-primary-arms-deluxe-extended-ar15.html