That depends on the definition you assign to "shoot". If it is how far the projectile travels before coming to rest on the ground or some other surface then the answer is at least several hundred yards. If we are talking about effective killing range then we need to know what type of shell we are shooting and what we are killing. 3 inch magnum game loads will kill dove out to maybe 20 yards in my experience. Maybe more maybe less depending on the gun and the shooter. Although I've never fired slugs from a .410 I doubt they would really do the job on a deer at 100 yards. I've not run the ballistics on them and I don't really see a need to. A .410 is basically an up close gun, even more so than other shotguns. They can do some unentended damage at range though so be safe.
Depends on the load. 30-300 yards
No. But there are revolvers made to shoot .410 shells that also fire .45 Colt cartridges,
None. While there are firearms made to shoot .45 Colt AND .410 shotshells, they are designed and made to do that. A regular .410 shotgun is NOT made to do that safely.
No.
You can shoot 45 Long Colt or .410 shot shells with this weapon. 45 ACP rounds will not work properly.
None.
No.
No it does not. The Browning BPS Hunter is a shotgun. The "410" signifies the shell size (guage) in this case .410
No made for 2and a half inch shells
Yes, they do.
Hell no. .410 is not a gauge it is a caliber. True gauge is a measurement of the number of lead balls of bore diameter that constitute a pound. The .243 bullet is a lot smaller than the diameter of a .410 barrel. There are some .410's that can shoot 45 colts but not all can.
No, not without barrel inserts.