All my research indicates that browning never offered the auto-5 shotgun in .410 gauge.
No
The book only shows 12 and 20 gauge.
Info available on Browning's site.
All my research which goes back to 1909 indicates that FN of belguim never produced a browning auto-5 shotgun in .410 gauge.
Yes
No
A semi-automatic shotgun made in 410, 28, 20 and 12 gauge.
Not as a cataloged item.
Not for commercial sale.
I don't know how long you want to go back, but Browning has made their Citori model in 410, 28, 20, 16, and 12 gauges for more than thirty years.
No, and no other company makes one either. 10,12,16,20,28, and 410 are what is being mfg at this time.