Not sure excactly what the question is. All Browning Sweet Sixteens (imported to Browning) after WWII are chambered for 2 3/4 16ga shells. The barrels are mared as such. Prior to 1941, Sweet 16's and Standard 16's were chambered for 2 9/16" shells.
Browning.com has sn tables you can look at.
Underside of the receiver forward of the shell carrier. sales@countrygunsmith.net
Not all 16ga Auto-5 shotguns are Sweet Sixteens. That is a specific lightweight model that was introduced in 1936. Chances are you have a standard weight Auto-5 in 16 gauge. Either way, it originally had a 2 9/16" chamber and is not suitable for modern 2 3/4" ammunition unless it has been modified for the longer shell. Value depends on condition. Anywhere from $350-$1000.
It will be marked on the barrel.
You need to see a gunsmith
The sales and sources of this shell vary day to day. Recommend that you do a search on google with the following key words: 16 ga 2 9/16" shotshells You will find several sources from there
you can't.
You need to take it to a gunsmith.
No it does not. The Browning BPS Hunter is a shotgun. The "410" signifies the shell size (guage) in this case .410
Should be marked on the barrel
From Browning's website - "allowing the shooter to retain shells in the magazine and eject a shell from the chamber without feeding from the magazine."
Brownings made prior to the Second World War could have been chambered in 16 ga, 2-9/16" length. That is no longer a standard SAAMI length and I know of no US companies who load it. You may be able to find a hand loader who can make them for you, you can investigate the European cartridge companies to see if there is a European equivalent, or you can have the gun converted to shoot the longer 2-3/4" shells.