According to the, interpolated from shipping records, serial number list in Charles Semmer's book, Remington Double Shotguns, 367004 would have been from 1906. There is no way to tell from the serial number or other markings on the gun whether it has Remington Steel or Damascus barrels. In a perfect world the Remington Steel barrels would be blued, while the Damascus barrels would show the Damascus pattern. However, in the last 105 years, many an unscrupulous seller has blued over Damascus barrels in a belief he/she could get more money for a steel barrel gun.
In this case these Model 1900 barrels from a 16-gauge KED-Grade are marked XKEDW down the bottoms --

and these KE-Grade 16-gauge barrels are marked XKEA1 down the bottom --

but this is not always the case!?!
No one can tell you over the internet what ammunition may or may not be safe in a given 100+ year old gun we haven't even seen pictures of. The gun needs to be evaluated by a qualified double gun smith (not Joe S**t the 870 parts replacer) with the gun in hand. Most Remington 12-gauge doubles were chambered for 2 5/8 inch shells and most 16-gauges for 2 9/16 inch shells, the "standard" shells of the time during which they were built.