I believe several of us attempted to be helpful John, are trying to be polite, and Researcher is apparently otherwise occupied and unavailable to clear up our confusion.
1. It's a Remington Hammerless Model of 1900; if with ejectors a KED. Researcher has documented a 1909 16g KE marked X K E A 1. Neither he nor Semmer know what that signifies.
http://www.doublegunshop.com/forums/ubbthreads.php?ubb=showflat&Number=418058#Post418058
2. There were 98,508 produced from 1900 to 1910; Serial numbers 300,000 to 398,508. SO the SN would suggest 1906, but I am not aware of a definitive list.
3. I suspect someone purposefully altered the 'D' to a 'I' and blued the Damascus barrels (and action which should be case colored) as to hide that fact. I could be wrong and maybe the KAI is original.
4. The buttstock was inexpertly replaced.
5. Mike A advised on the first page that the gun needs to be evaluated by a doublegun smith
6. The /// should be accompanied by a pair of 1 or 2 digit numbers, signifying 3__ pellets patterned using 1 1/4 ounces of #8 with 511 pellets to the load in a 30” circle at 40 yards. If stamped 33 that would be 333/511 = 65%. IF the 4 means 304 = 59.5% = Modified
7. What can you add, other than an estimate of value?