The records for the Model 1 do not exist, they best you can generally get is from 1873 - 1894 can narrow that down but then only estimates is based on features.
I don't claim to be a Model 1 Rolling Block expert, although I would love to own one ESPECIALLY in 22 (that's all I collect), but I don't ever recall seeing one with a H prefix on the serial no.
You will have to contact Remington.
your Remington rand was made in 1917.
Your Remington model 1903-A3 was made in the year 1942.It was chambered in the caliber 30-06.
That's a very late gun probably 1933, see related links for date code stamp.
1 in 10
The Remington model 4 rolling block rifle was made between 1890-1933.If you have a model 4S military rifle,then these were made by Remington from 1913-1923.your rifle has either a 22.5in barrel,or a 24in barrel.These were chambered in .22,.25 Stevens,or 32 caliber.
Early 1911
Manufactured 1909
List the caliber and then someone will try to help you.
Impossible to tell unless you have the factory records.
Model 581.
$500 up