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
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.
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."