It's made in 1925, it's late enough that it should have the date code stamped on the left side of the barrel in front of receiver, check Remington Society of America for details.
Model 581.
First year 1909
1919 depends entirely on condition $100-1,200
Your Remington model 1903-A3 was made in the year 1942.It was chambered in the caliber 30-06.
You will have to contact Remington.
1 in 10
Impossible to tell unless you have the factory records.
List the caliber and then someone will try to help you.
Are you sure of the model number?
$500 up
SN's were not required before 1968
1920