Burris does not provide the country of origin for their AR-P.E.P.R Tactical Rifle Scope Rings with Mount. I'd recommend reaching out to them directly for more information.
Yes, the Burris AR-P.E.P.R Rifle Scope Mount will be compatible with the ADE 3-12x40mm Rifle scope.
The 30mm diameter Burris AR-P.E.P.R Rifle Scope Mount will work with that scope.
The Burris AR-P.E.P.R Rifle Scope Mount should work with a scope with a 50mm objective lens. However, to be sure you would need to the know the outside diameter of the objective lens housing. The guide below will help you determine the height you need.
https://www.opticsplanet.com/howto/how-to-how-to-measure-scope-height-scope-ring-height.html
Yes, the Burris AR-P.E.P.R Rifle Scope Mount is specifically designed for mounting to AR platform rifles flattop rails. The 30mm model will accommodate your Sig Sauer Tango MSR LPVO and function as intended.
Yes, the Burris AR-P.E.P.R Tactical Rifle Scope Mount MPN:410341 can accommodate a 44mm objective lens without issue. It has a centerline height of 1.60".
The dimensions of the Burris AR-P.E.P.R Tactical Rifle Scope Rings with Mount can be seen in the last product image on the left side of the product page.
The distance between the Burris AR-P.E.P.R Tactical Rifle Scope Rings is 2.15" for all the models except the 34mm diameter model.
The Burris AR-P.E.P.R Tactical Rifle Scope Rings with Mount will probably not work with the Barska 3-9x42 compact scope. The space between the mounting rings on the Burris AR-P.E.P.R Tactical Rifle Scope Rings with Mount will be too far apart. I recommend using a set up individual 1" scope rings for that scope.
The Burris Eliminator III LaserScope 4-16x50mm has an integrated mounting platform in the bottom of the scope and the included mount clamps. So the Burris Eliminator III LaserScope 4-16x50 mm will not work with this Burris AR-P.E.P.R Tactical Rifle Scope Rings with Mount. The Burris Eliminator III LaserScope will mount directly to the picatinny mount on that rifle using the Burris Eliminator III LaserScope's supplied mounting clamps.